{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0001","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60cee0822e048edb940546216e47336ffa85af534564cad3a799f9a61ba080c1","family_id":"qf-density15-selection-order-factor-triple","seed":202610156001,"semantic_template":"qf-density15-selection-order-factor-triple/[\"1\",\"1\",\"1\",true]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=0, k=0, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(1, 1, 1, Yes)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (1, 1, 1, Yes)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0002","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef5742301ec06bd4e42217a9868a77f7e7fb51753b085ea67ba15a7bce4b7e3e","family_id":"qf-density15-selection-order-factor-triple","seed":202610156002,"semantic_template":"qf-density15-selection-order-factor-triple/[\"1\",\"1\",\"1\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=1, k=1, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(1, 1, 1, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (1, 1, 1, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0003","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea448859106ceebfb01660a22971587895598f3783231ef5f2ce61042e07b2d8","family_id":"qf-density15-selection-order-factor-triple","seed":202610156003,"semantic_template":"qf-density15-selection-order-factor-triple/[\"2432902008176640000\",\"20\",\"121645100408832000\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=20, k=19, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(2432902008176640000, 20, 121645100408832000, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (2432902008176640000, 20, 121645100408832000, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0004","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6cadb0f8c5fc0741d2255539aebe2e8a0f2a1fa65494506aedb41c15903a1b2","family_id":"qf-density15-selection-order-factor-triple","seed":202610156004,"semantic_template":"qf-density15-selection-order-factor-triple/[\"24\",\"4\",\"6\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=4, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(24, 4, 6, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (24, 4, 6, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0005","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fba982f4272dd37c4b9292e28264ab3259e4ac678ef5242baf5019b3df1d5f0","family_id":"qf-density15-selection-order-factor-triple","seed":202610156005,"semantic_template":"qf-density15-selection-order-factor-triple/[\"20\",\"10\",\"2\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=5, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(20, 10, 2, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (20, 10, 2, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0006","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee3bdf1d6296b03b5cb1753854842bd6729f1e039fbc67a461e8e133cbf61a95","family_id":"qf-density15-selection-order-factor-triple","seed":202610156006,"semantic_template":"qf-density15-selection-order-factor-triple/[\"120\",\"20\",\"6\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=6, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(120, 20, 6, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (120, 20, 6, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0007","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdd65d03f3c8216fd4df39deda630fade294b474c9e2c2a246d0ffc0e68ef101","family_id":"qf-density15-selection-order-factor-triple","seed":202610156007,"semantic_template":"qf-density15-selection-order-factor-triple/[\"42\",\"21\",\"2\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=7, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(42, 21, 2, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (42, 21, 2, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0008","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"228955c615959b2648744bfdc3636ec0af83dce8d6cc077051d703d36aae2ee3","family_id":"qf-density15-selection-order-factor-triple","seed":202610156008,"semantic_template":"qf-density15-selection-order-factor-triple/[\"336\",\"56\",\"6\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=8, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(336, 56, 6, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (336, 56, 6, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0009","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"255955b837458c22f9d9c2f2d7a2f4dbd56b84e3ba8031edec75b6e0fa83fe0e","family_id":"qf-density15-selection-order-factor-triple","seed":202610156009,"semantic_template":"qf-density15-selection-order-factor-triple/[\"72\",\"36\",\"2\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=9, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(72, 36, 2, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (72, 36, 2, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0010","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9e243de6ec52ebff4ea222a3b24d81432da67601498d8d79ef95f9b2dd9c74f","family_id":"qf-density15-selection-order-factor-triple","seed":202610156010,"semantic_template":"qf-density15-selection-order-factor-triple/[\"720\",\"120\",\"6\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=10, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(720, 120, 6, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (720, 120, 6, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0011","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaa43e2dceaad033f2f60063754fc41c7509a3b2ec2cadbe69b5aa1b150e768b","family_id":"qf-density15-selection-order-factor-triple","seed":202610156011,"semantic_template":"qf-density15-selection-order-factor-triple/[\"110\",\"55\",\"2\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=11, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(110, 55, 2, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (110, 55, 2, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0012","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f814edab0a2c15a9bec58c7ed6aff4e98de60f55459ee33d4211dbf5a5781e6d","family_id":"qf-density15-selection-order-factor-triple","seed":202610156012,"semantic_template":"qf-density15-selection-order-factor-triple/[\"1320\",\"220\",\"6\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=12, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(1320, 220, 6, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (1320, 220, 6, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0013","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05e5f6c708aadb80caa6a8fd38e03324e0ace849aef03f41dd02b69c01014100","family_id":"qf-density15-selection-order-factor-triple","seed":202610156013,"semantic_template":"qf-density15-selection-order-factor-triple/[\"156\",\"78\",\"2\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=13, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(156, 78, 2, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (156, 78, 2, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0014","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5185c93289627f4d7bf9fb6609298203d52423cf2a42bd126baaceb51ac9e0e7","family_id":"qf-density15-selection-order-factor-triple","seed":202610156014,"semantic_template":"qf-density15-selection-order-factor-triple/[\"2184\",\"364\",\"6\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=14, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(2184, 364, 6, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (2184, 364, 6, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0015","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a736e72f8062d60b23b38a3290ad8573661f0458026e8978dd7dc11085c9b1b","family_id":"qf-density15-selection-order-factor-triple","seed":202610156015,"semantic_template":"qf-density15-selection-order-factor-triple/[\"210\",\"105\",\"2\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=15, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(210, 105, 2, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (210, 105, 2, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0016","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41f89fa8720296e21d49e7a958aa1cd7e117c3abe1d143ab1cf4c717cd26145b","family_id":"qf-density15-selection-order-factor-triple","seed":202610156016,"semantic_template":"qf-density15-selection-order-factor-triple/[\"3360\",\"560\",\"6\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=16, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(3360, 560, 6, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (3360, 560, 6, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0017","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e398fbe1a8f7f943bbca8d8ad1ce7098af5127a30cf893a8ccf88290c6815a5e","family_id":"qf-density15-selection-order-factor-triple","seed":202610156017,"semantic_template":"qf-density15-selection-order-factor-triple/[\"272\",\"136\",\"2\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=17, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(272, 136, 2, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (272, 136, 2, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0018","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddaf368f6b7b87d27af68a05e0e69c079070d08ed40a9720b309ab82d7c83d15","family_id":"qf-density15-selection-order-factor-triple","seed":202610156018,"semantic_template":"qf-density15-selection-order-factor-triple/[\"4896\",\"816\",\"6\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=18, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(4896, 816, 6, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (4896, 816, 6, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0019","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ede66f587ec1ff32a5a5386228db5722a24b1117ab293dc573212652687d71e9","family_id":"qf-density15-selection-order-factor-triple","seed":202610156019,"semantic_template":"qf-density15-selection-order-factor-triple/[\"342\",\"171\",\"2\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=19, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(342, 171, 2, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (342, 171, 2, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0020","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"937c154b305b47c33952cc1ce6f3dd2e220750fa511619535d8f85e05b253d78","family_id":"qf-density15-selection-order-factor-triple","seed":202610156020,"semantic_template":"qf-density15-selection-order-factor-triple/[\"6840\",\"1140\",\"6\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=20, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(6840, 1140, 6, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (6840, 1140, 6, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0021","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bc30a995383926c58515847683062f15e44b326cb4ff45959ab8ff5d9e46c0b","family_id":"qf-density15-selection-order-factor-triple","seed":202610156021,"semantic_template":"qf-density15-selection-order-factor-triple/[\"420\",\"210\",\"2\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=21, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(420, 210, 2, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (420, 210, 2, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0022","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bbf236b88071146162a8814a471c78544b6b3ba96fd735a97e35195b31d9129","family_id":"qf-density15-selection-order-factor-triple","seed":202610156022,"semantic_template":"qf-density15-selection-order-factor-triple/[\"9240\",\"1540\",\"6\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=22, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(9240, 1540, 6, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (9240, 1540, 6, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0023","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a66a5256ebe697f240f07d4bc1ede3ed55253548d113a30b4848b17f397d83dd","family_id":"qf-density15-selection-order-factor-triple","seed":202610156023,"semantic_template":"qf-density15-selection-order-factor-triple/[\"506\",\"253\",\"2\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=23, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(506, 253, 2, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (506, 253, 2, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0024","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ca3833fab6aedce1afbf663b5cf9bfc7a238503ca2cc07d13f533cd36663c81","family_id":"qf-density15-selection-order-factor-triple","seed":202610156024,"semantic_template":"qf-density15-selection-order-factor-triple/[\"12144\",\"2024\",\"6\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=24, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(12144, 2024, 6, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (12144, 2024, 6, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0025","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7af213e3805a6bccd0e71730808e1d66490de636aea803c8a9480fa45d2114ed","family_id":"qf-density15-selection-order-factor-triple","seed":202610156025,"semantic_template":"qf-density15-selection-order-factor-triple/[\"600\",\"300\",\"2\",false]"},"fields":{"title":"Ordered/Unordered Counts, Factorial, and Empty Selection","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=25, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).","answer":{"kind":"expression","value":"(600, 300, 2, No)","explanation":"Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (600, 300, 2, No)."},"skills":["Ordered/Unordered Counts, Factorial, and Empty Selection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0026","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73e41c801eb0cbb718ab8afdb1aa99a162a9fbde31048d906fc4e61149b51a16","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156026,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[]],[false,false,true,\"invalid\"],[false,false,false,\"invalid\"]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(0)=∅, k=0, candidates=[[], [0], [0, 0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, No, No], [Yes, No, No], [Yes, Yes, No], [Yes, No, No], [[], invalid-selection, invalid-selection], [[], invalid-selection, invalid-selection])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, No, No], [Yes, No, No], [Yes, Yes, No], [Yes, No, No], [[], invalid-selection, invalid-selection], [[], invalid-selection, invalid-selection])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0027","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b8aef7843be0e95bbc6b2c80d3c5e2216d878b6a1a2afab0dd72da9a985c6a2","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156027,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[1,0]],[true,true,true,[0,1]],[true,true,false,\"invalid\"]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(4)={0,…,3}, k=2, candidates=[[1, 0], [0, 1], [0, 0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, No], [Yes, Yes, No], [[1, 0], [0, 1], invalid-selection], [[0, 1], [0, 1], invalid-selection])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, No], [Yes, Yes, No], [[1, 0], [0, 1], invalid-selection], [[0, 1], [0, 1], invalid-selection])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0028","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27a2de5a8de95630537e7a3c112411a9a3a1386ad0e315686143bf1fd5f31fd9","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156028,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[false,true,true,\"invalid\"],[true,false,true,\"invalid\"]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(4)={0,…,3}, k=2, candidates=[[0], [0, 1, 2], [0, 4]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([No, No, Yes], [Yes, Yes, No], [Yes, Yes, Yes], [No, No, No], [invalid-selection, invalid-selection, invalid-selection], [invalid-selection, invalid-selection, invalid-selection])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, No, Yes], [Yes, Yes, No], [Yes, Yes, Yes], [No, No, No], [invalid-selection, invalid-selection, invalid-selection], [invalid-selection, invalid-selection, invalid-selection])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0029","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e0b3dc00094fc11c573233296a51f431b5f852437439249d34fe57d8f731621","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156029,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[0,1]],[true,true,true,[0,1]],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 1], [0, 1], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [[0, 1], [0, 1], [0, 1]], [[0, 1], [0, 1], [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [[0, 1], [0, 1], [0, 1]], [[0, 1], [0, 1], [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0030","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6956d4501a14ad93da420ee2a6ed2dc85bd0b7f1c81286ea234cca70b62eaf66","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156030,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[true,true,true,[0,1]],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0, 1], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [invalid-selection, [0, 1], [0, 1]], [invalid-selection, [0, 1], [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [invalid-selection, [0, 1], [0, 1]], [invalid-selection, [0, 1], [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0031","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"343668673d703b87ddc96e55b10e2a58e5bb7bbc583f7e13fda8361ba0647546","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156031,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,false,\"invalid\"],[true,true,true,[0,1]],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 0], [0, 1], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [No, Yes, Yes], [invalid-selection, [0, 1], [0, 1]], [invalid-selection, [0, 1], [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [No, Yes, Yes], [invalid-selection, [0, 1], [0, 1]], [invalid-selection, [0, 1], [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0032","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16c977a93adb5ce6e78973ed9108f0e95d611a5df176fa033fb3b80247450271","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156032,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,false,true,\"invalid\"],[true,true,true,[0,1]],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 8], [0, 1], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [No, Yes, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [invalid-selection, [0, 1], [0, 1]], [invalid-selection, [0, 1], [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, Yes, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [invalid-selection, [0, 1], [0, 1]], [invalid-selection, [0, 1], [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0033","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90dae1be18dd72311e7680968675452879c6569f96dfc56ffef3b76ca95fed65","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156033,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[0,1]],[false,true,true,\"invalid\"],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 1], [0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, No, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [[0, 1], invalid-selection, [0, 1]], [[0, 1], invalid-selection, [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, No, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [[0, 1], invalid-selection, [0, 1]], [[0, 1], invalid-selection, [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0034","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c74b0c48fe876e0c13e207036e839b4a9de46ecf32f00979034af4663d906102","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156034,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[false,true,true,\"invalid\"],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([No, No, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, No, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0035","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4bb8a6a7858e65cbc3dbc2fb3d1f8bb7a11371ff84e436c4fd7b81f82d469cf","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156035,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,false,\"invalid\"],[false,true,true,\"invalid\"],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 0], [0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, No, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, No, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0036","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"041bcc2bb24ad70d05fc3dc082f1a61808b2652a637864e6ed05b7cb5b4eb183","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156036,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,false,true,\"invalid\"],[false,true,true,\"invalid\"],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 8], [0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, No, Yes], [No, Yes, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, No, Yes], [No, Yes, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0037","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd46746816abc702b1a5555547af17799372c84d2505b5d943cc98988fb20f25","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156037,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[0,1]],[true,true,false,\"invalid\"],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 1], [0, 0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [Yes, No, Yes], [[0, 1], invalid-selection, [0, 1]], [[0, 1], invalid-selection, [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [Yes, No, Yes], [[0, 1], invalid-selection, [0, 1]], [[0, 1], invalid-selection, [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0038","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe7ceaab207293c0ffd7f22d0cfc82a3870f3834f8a40bf979846d760cfcee5b","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156038,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[true,true,false,\"invalid\"],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0, 0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0039","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b3edd7cd2a478b7a01eff2b2203c002aa1e37bd18bb93d2b0f28129408c225a","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156039,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,false,\"invalid\"],[true,true,false,\"invalid\"],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 0], [0, 0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0040","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9452dc7beeddf235aec20134d8adf675fd9153dd0cdf1e253d4c6ddae6349684","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156040,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,false,true,\"invalid\"],[true,true,false,\"invalid\"],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 8], [0, 0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [No, Yes, Yes], [Yes, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, Yes, Yes], [Yes, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0041","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d2d75bc644e066bd7f28a9f4e3117c6e401c60a72f6a55158144bc6093f4ed2","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156041,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[0,1]],[true,false,true,\"invalid\"],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 1], [0, 8], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, No, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [[0, 1], invalid-selection, [0, 1]], [[0, 1], invalid-selection, [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, No, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [[0, 1], invalid-selection, [0, 1]], [[0, 1], invalid-selection, [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0042","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"719fcef4c6cc70f5d51dd0bec10f20b264a9c9d0b8288c9478af07e7c3cc9aee","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156042,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[true,false,true,\"invalid\"],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0, 8], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([No, Yes, Yes], [Yes, No, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, Yes, Yes], [Yes, No, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0043","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ea01819db1e53323116cc0f5bbbd8748ab5bcbba3f149c05d240282a72ab766","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156043,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,false,\"invalid\"],[true,false,true,\"invalid\"],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 0], [0, 8], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, No, Yes], [No, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, No, Yes], [No, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0044","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d96bbd458f6455ccdd20d1373d8aed9c4a109050499600868b8a9bf2aa833bf6","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156044,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,false,true,\"invalid\"],[true,false,true,\"invalid\"],[true,true,true,[0,1]]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 8], [0, 8], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [No, No, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, No, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0045","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec466a1af4ed2fadf81251eb0681e7e987ed86d3a460b4aae6db80d7d5a5757e","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156045,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[0,1]],[true,true,true,[0,1]],[false,true,true,\"invalid\"]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 1], [0, 1], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, No], [[0, 1], [0, 1], invalid-selection], [[0, 1], [0, 1], invalid-selection])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, No], [[0, 1], [0, 1], invalid-selection], [[0, 1], [0, 1], invalid-selection])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0046","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"339f41b91beb3a211af71e3fc8c7dc9ba570bcbd4841f5675928ca086673f1d7","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156046,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[true,true,true,[0,1]],[false,true,true,\"invalid\"]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0, 1], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([No, Yes, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, Yes, No], [invalid-selection, [0, 1], invalid-selection], [invalid-selection, [0, 1], invalid-selection])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, Yes, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, Yes, No], [invalid-selection, [0, 1], invalid-selection], [invalid-selection, [0, 1], invalid-selection])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0047","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8d27230a7bc753c2c6db50bfff09f52ee3a4322ddf32f5b0262c5ab84b1a779","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156047,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,false,\"invalid\"],[true,true,true,[0,1]],[false,true,true,\"invalid\"]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 0], [0, 1], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, No], [Yes, Yes, Yes], [No, Yes, Yes], [No, Yes, No], [invalid-selection, [0, 1], invalid-selection], [invalid-selection, [0, 1], invalid-selection])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, No], [Yes, Yes, Yes], [No, Yes, Yes], [No, Yes, No], [invalid-selection, [0, 1], invalid-selection], [invalid-selection, [0, 1], invalid-selection])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0048","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"761e9d2cfc40d29f9c1d7e1f40a043803695f42eec8327ba9438f418f57166b5","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156048,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,false,true,\"invalid\"],[true,true,true,[0,1]],[false,true,true,\"invalid\"]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 8], [0, 1], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, Yes, No], [No, Yes, Yes], [Yes, Yes, Yes], [No, Yes, No], [invalid-selection, [0, 1], invalid-selection], [invalid-selection, [0, 1], invalid-selection])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, No], [No, Yes, Yes], [Yes, Yes, Yes], [No, Yes, No], [invalid-selection, [0, 1], invalid-selection], [invalid-selection, [0, 1], invalid-selection])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0049","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ab4d0c1127f383121d82885ad83b77af3a139aa281cc57ff4f84b4c67f88a8b","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156049,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[0,1]],[false,true,true,\"invalid\"],[false,true,true,\"invalid\"]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 1], [0], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([Yes, No, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, No], [[0, 1], invalid-selection, invalid-selection], [[0, 1], invalid-selection, invalid-selection])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, No, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, No], [[0, 1], invalid-selection, invalid-selection], [[0, 1], invalid-selection, invalid-selection])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0050","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e51dd995f68930564387b8ad9630c520f9fd7f2847349a7d8dcf6af18d690316","family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156050,"semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[false,true,true,\"invalid\"],[false,true,true,\"invalid\"]]"},"fields":{"title":"Candidate Selection Length, Typing, and Distinctness","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.","answer":{"kind":"expression","value":"([No, No, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, No], [invalid-selection, invalid-selection, invalid-selection], [invalid-selection, invalid-selection, invalid-selection])","explanation":"Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, No, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, No], [invalid-selection, invalid-selection, invalid-selection], [invalid-selection, invalid-selection, invalid-selection])."},"skills":["Candidate Selection Length, Typing, and Distinctness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0051","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd9f52e13b40888976278b3a6c199f33af050a9ceec30dbe9d2fb9583dd8a4a2","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156051,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"1\",\"0\",\"1\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=1≥1, k=0, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(1, 0, 1, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (1, 0, 1, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0052","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"076f9a17168e50f9d6ed42a3f611e500703ac26ce2274b4a793e30233e30f1fa","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156052,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"1\",\"1\",\"0\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=3≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(1, 1, 0, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (1, 1, 0, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0053","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11a263caa4ed466e5d68682a20cd45e23c8f846fcc78b468cec2fbb2bd241104","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156053,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"184756\",\"92378\",\"92378\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=20≥1, k=10, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(184756, 92378, 92378, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (184756, 92378, 92378, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0054","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae4631ca62bc8fd39b3767cce472ecd96ab0b1f97417fdfd6438449a1f20ca7c","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156054,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"6\",\"5\",\"1\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=6≥1, k=5, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(6, 5, 1, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (6, 5, 1, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0055","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf82e963352cb7a48777c6eb58fe8b023e0119b511f4b1fecd4528ddb6e3f216","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156055,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"10\",\"4\",\"6\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=5≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(10, 4, 6, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (10, 4, 6, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0056","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4abaff226b29c4c22270b8ee98e1f5ebc6c71de6b2b102c4671522ae2d445579","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156056,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"20\",\"10\",\"10\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=6≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(20, 10, 10, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (20, 10, 10, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0057","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c5a9ebbb8b99522320c2ab28dc39ae87bf06f4cfdfcbfadf600b0352c03c38e","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156057,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"21\",\"6\",\"15\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=7≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(21, 6, 15, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (21, 6, 15, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0058","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"228d6c3a0c087405cbb0e70be803b64241ea9d199e3c9bbb037da14d2f7ec699","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156058,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"56\",\"21\",\"35\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=8≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(56, 21, 35, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (56, 21, 35, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0059","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57eb6cdbfcd3010f8267c8299f86b2ef9db08792b528c25068b16c93c109e618","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156059,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"36\",\"8\",\"28\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=9≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(36, 8, 28, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (36, 8, 28, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0060","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de5b9191eac241fb5a65cffcbfd6695181cd8f8aa145e48b6df98efc6bd2ea4d","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156060,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"120\",\"36\",\"84\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=10≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(120, 36, 84, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (120, 36, 84, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0061","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bf93d2ef7c470a08572b02b5249a13cc53c52e87cb9f366db4673e862cd6a1c","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156061,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"55\",\"10\",\"45\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=11≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(55, 10, 45, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (55, 10, 45, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0062","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"956df692cff1633dcf643c1b0c710144c05ef4fd1ba6993eac2f44d419dc6e84","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156062,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"220\",\"55\",\"165\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=12≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(220, 55, 165, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (220, 55, 165, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0063","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f63ca93665dd87327baa50ae18921fdecd99d9b3499105bf0036b39f966b36ba","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156063,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"78\",\"12\",\"66\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=13≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(78, 12, 66, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (78, 12, 66, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0064","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"771bff407b97f01b6d4306c1c8315ebd2e8917ffcb58ba825fac56145dadd9f6","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156064,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"364\",\"78\",\"286\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=14≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(364, 78, 286, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (364, 78, 286, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0065","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"438d0cc551595bba202bd36572805458c41c6de8741f941ebd596d404044702d","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156065,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"105\",\"14\",\"91\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=15≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(105, 14, 91, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (105, 14, 91, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0066","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78ef54a73296052f3c7382e6713f6aa8cc7b59ade6e508b29c836a9973748852","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156066,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"560\",\"105\",\"455\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=16≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(560, 105, 455, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (560, 105, 455, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0067","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b313e1f4fd7b476fdb702d8fa17197e744f8ef6a967e0842df479debd77291f9","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156067,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"136\",\"16\",\"120\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=17≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(136, 16, 120, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (136, 16, 120, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0068","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee2b31136abcda1877e012cd7b4142dea03bba144b50bf854f647340e0cc836e","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156068,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"816\",\"136\",\"680\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=18≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(816, 136, 680, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (816, 136, 680, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0069","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55c8a482242218f744a4b6d07fefb843b99701f8687e5b13182ed95f1565b6e1","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156069,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"171\",\"18\",\"153\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=19≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(171, 18, 153, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (171, 18, 153, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0070","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffc6f1a77b45a7bb3b5cccaeedd0fc2fbc139a96c40b279ecc117e6ff1d5d69e","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156070,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"1140\",\"171\",\"969\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=20≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(1140, 171, 969, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (1140, 171, 969, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0071","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bae398eec6a0cc66858a2510315d80233602d63d14c1b8713e66f6d6780833c0","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156071,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"210\",\"20\",\"190\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=21≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(210, 20, 190, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (210, 20, 190, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0072","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb2ea6ddb84cc6ba728be531406149358f9fdafe9875c67a0cce6ba2957cca43","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156072,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"1540\",\"210\",\"1330\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=22≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(1540, 210, 1330, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (1540, 210, 1330, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0073","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f2a741603e634d53a52fa55dceff990b93d451e154eb7e3d998bd66344994ce","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156073,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"253\",\"22\",\"231\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=23≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(253, 22, 231, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (253, 22, 231, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0074","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b13204cc84041339a65975a0b06f532cbdc1745cb33ba5a2aa07e5877c8f31db","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156074,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"2024\",\"253\",\"1771\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=24≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(2024, 253, 1771, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (2024, 253, 1771, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0075","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adf80306eb329e2e3be4235bb1588fcc76a665e3935ad423c76c40577cc1c2eb","family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156075,"semantic_template":"qf-density15-selection-one-distinguished-counts/[\"300\",\"24\",\"276\"]"},"fields":{"title":"Combinations Including or Excluding a Fixed Existing Label","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=25≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).","answer":{"kind":"expression","value":"(300, 24, 276, 0)","explanation":"Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (300, 24, 276, 0)."},"skills":["Combinations Including or Excluding a Fixed Existing Label"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0076","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bc2137a06b26378b2e959ceb9399f6983e198fe368f91261da814adee1626ca","family_id":"qf-density15-selection-forced-initial-order","seed":202610156076,"semantic_template":"qf-density15-selection-forced-initial-order/[0,0,\"1\",\"1\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=0, k=0, r=0; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(0, 0, 1, 1, 1/1)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (0, 0, 1, 1, 1/1)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0077","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a291d277e637f46d3040a9356537e41b6adec061f352d443db11637e671abed0","family_id":"qf-density15-selection-forced-initial-order","seed":202610156077,"semantic_template":"qf-density15-selection-forced-initial-order/[4,2,\"12\",\"12\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=4, k=2, r=0; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(4, 2, 12, 12, 1/1)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (4, 2, 12, 12, 1/1)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0078","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf379b32ecbc8d8670ee8fc8b6aa9beb67bafea5386229a3bbe86670ee6e1c42","family_id":"qf-density15-selection-forced-initial-order","seed":202610156078,"semantic_template":"qf-density15-selection-forced-initial-order/[2,0,\"1\",\"12\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=4, k=2, r=2; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(2, 0, 1, 12, 1/12)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (2, 0, 1, 12, 1/12)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0079","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29c987a6fe7e864ae9330dc96522934f6b1fd311f9f374009cb6911f4c745577","family_id":"qf-density15-selection-forced-initial-order","seed":202610156079,"semantic_template":"qf-density15-selection-forced-initial-order/[16,16,\"20922789888000\",\"263130836933693530167218012160000000\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=32, k=32, r=16; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(16, 16, 20922789888000, 263130836933693530167218012160000000, 1/12576278705767096320000)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (16, 16, 20922789888000, 263130836933693530167218012160000000, 1/12576278705767096320000)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0080","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c2ea98abd65da0ab0687c56926c5a6e6c79cc4a01efdd679c2d1bf626bb5864","family_id":"qf-density15-selection-forced-initial-order","seed":202610156080,"semantic_template":"qf-density15-selection-forced-initial-order/[4,2,\"12\",\"60\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=5, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(4, 2, 12, 60, 1/5)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (4, 2, 12, 60, 1/5)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0081","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5b7d4c864864a1cb25afbbb49d3faa7a1856577c1bbbf8b76197d4932277c32","family_id":"qf-density15-selection-forced-initial-order","seed":202610156081,"semantic_template":"qf-density15-selection-forced-initial-order/[5,2,\"20\",\"120\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=6, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(5, 2, 20, 120, 1/6)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (5, 2, 20, 120, 1/6)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0082","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0c6de44f5a94e4cf652ae20b2454ad58bf2542757eae12e9966fb66e3198ef7","family_id":"qf-density15-selection-forced-initial-order","seed":202610156082,"semantic_template":"qf-density15-selection-forced-initial-order/[6,2,\"30\",\"210\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=7, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(6, 2, 30, 210, 1/7)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (6, 2, 30, 210, 1/7)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0083","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"460322fbe19e013c8265376e4749b8ca6a74922e7cc853d7ca173730e631536f","family_id":"qf-density15-selection-forced-initial-order","seed":202610156083,"semantic_template":"qf-density15-selection-forced-initial-order/[7,2,\"42\",\"336\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=8, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(7, 2, 42, 336, 1/8)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (7, 2, 42, 336, 1/8)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0084","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50856fe109d844d03c4c58ebbc51a57b96592f813a38d17ca8fdbfbb255c9ba5","family_id":"qf-density15-selection-forced-initial-order","seed":202610156084,"semantic_template":"qf-density15-selection-forced-initial-order/[8,2,\"56\",\"504\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=9, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(8, 2, 56, 504, 1/9)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (8, 2, 56, 504, 1/9)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0085","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5a78a936355f3e1c425a06f56690462f46df9bff0933c678820a5481dc244b4","family_id":"qf-density15-selection-forced-initial-order","seed":202610156085,"semantic_template":"qf-density15-selection-forced-initial-order/[9,2,\"72\",\"720\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=10, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(9, 2, 72, 720, 1/10)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (9, 2, 72, 720, 1/10)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0086","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dd3e8d66a5518d8a207ce96bda995a14b8313a4d46e7b0f9a8d4cfef8bcba00","family_id":"qf-density15-selection-forced-initial-order","seed":202610156086,"semantic_template":"qf-density15-selection-forced-initial-order/[10,2,\"90\",\"990\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=11, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(10, 2, 90, 990, 1/11)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (10, 2, 90, 990, 1/11)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0087","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78e533d9c198bb90ff6dfa4a2829024d6c87c6d16ef770c3a860e1f067dffe31","family_id":"qf-density15-selection-forced-initial-order","seed":202610156087,"semantic_template":"qf-density15-selection-forced-initial-order/[11,2,\"110\",\"1320\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=12, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(11, 2, 110, 1320, 1/12)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (11, 2, 110, 1320, 1/12)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0088","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39c4ea34a4d6674932e75b40f8764f459e090b2a8493dff46b587497a2e76437","family_id":"qf-density15-selection-forced-initial-order","seed":202610156088,"semantic_template":"qf-density15-selection-forced-initial-order/[12,2,\"132\",\"1716\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=13, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(12, 2, 132, 1716, 1/13)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (12, 2, 132, 1716, 1/13)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0089","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fde65df59ba617d541d1aaf7bd6727227cdfcd9633a636cf525bf4ffa402da16","family_id":"qf-density15-selection-forced-initial-order","seed":202610156089,"semantic_template":"qf-density15-selection-forced-initial-order/[13,2,\"156\",\"2184\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=14, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(13, 2, 156, 2184, 1/14)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (13, 2, 156, 2184, 1/14)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0090","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1623e6169e1c7259a6c6781a8deec8b9c235d0b265afd2e677a755cc782ba604","family_id":"qf-density15-selection-forced-initial-order","seed":202610156090,"semantic_template":"qf-density15-selection-forced-initial-order/[14,2,\"182\",\"2730\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=15, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(14, 2, 182, 2730, 1/15)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (14, 2, 182, 2730, 1/15)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0091","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3d969a12c270afb1180657e5f2fa6f9e836fd5867a1ebebe32d44c292304b0a","family_id":"qf-density15-selection-forced-initial-order","seed":202610156091,"semantic_template":"qf-density15-selection-forced-initial-order/[15,2,\"210\",\"3360\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=16, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(15, 2, 210, 3360, 1/16)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (15, 2, 210, 3360, 1/16)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0092","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ca46d6acf9b60e49394d198a6434b49bfedc21464c46fcc0f516fc045ac3b5d","family_id":"qf-density15-selection-forced-initial-order","seed":202610156092,"semantic_template":"qf-density15-selection-forced-initial-order/[16,2,\"240\",\"4080\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=17, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(16, 2, 240, 4080, 1/17)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (16, 2, 240, 4080, 1/17)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0093","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3ea88902c31e2b3a7d43a02c8923d0767f9a42edba3c493d84dfdb8d95a2c71","family_id":"qf-density15-selection-forced-initial-order","seed":202610156093,"semantic_template":"qf-density15-selection-forced-initial-order/[17,2,\"272\",\"4896\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=18, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(17, 2, 272, 4896, 1/18)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (17, 2, 272, 4896, 1/18)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0094","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1052287bbcd2954ffd5f4876a802b8a0801fecbda92e768bcab4a8768e6458c","family_id":"qf-density15-selection-forced-initial-order","seed":202610156094,"semantic_template":"qf-density15-selection-forced-initial-order/[18,2,\"306\",\"5814\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=19, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(18, 2, 306, 5814, 1/19)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (18, 2, 306, 5814, 1/19)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0095","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f279299263277ba30c3887332d00052e76bca206d1d78c12a8f029004455fc8f","family_id":"qf-density15-selection-forced-initial-order","seed":202610156095,"semantic_template":"qf-density15-selection-forced-initial-order/[19,2,\"342\",\"6840\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=20, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(19, 2, 342, 6840, 1/20)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (19, 2, 342, 6840, 1/20)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0096","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca0499b81a025a0238d8eb095121c66d1263bdf7821ed6019c5d9ae7a61e669f","family_id":"qf-density15-selection-forced-initial-order","seed":202610156096,"semantic_template":"qf-density15-selection-forced-initial-order/[20,2,\"380\",\"7980\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=21, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(20, 2, 380, 7980, 1/21)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (20, 2, 380, 7980, 1/21)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0097","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"417f9c41bc5c4ddae5c89c0b8971749b25a5306ccd4380b71909fbf2237a28d4","family_id":"qf-density15-selection-forced-initial-order","seed":202610156097,"semantic_template":"qf-density15-selection-forced-initial-order/[21,2,\"420\",\"9240\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=22, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(21, 2, 420, 9240, 1/22)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (21, 2, 420, 9240, 1/22)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0098","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"067c2fde9c6e67c0aa3e13496eb2c1fb847b94548fb4858e4bf3bad36a7b4287","family_id":"qf-density15-selection-forced-initial-order","seed":202610156098,"semantic_template":"qf-density15-selection-forced-initial-order/[22,2,\"462\",\"10626\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=23, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(22, 2, 462, 10626, 1/23)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (22, 2, 462, 10626, 1/23)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0099","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e65515ae1a5837c5a848119a3793901ce4d8ae8c08ff0b198a6ce0c40372279","family_id":"qf-density15-selection-forced-initial-order","seed":202610156099,"semantic_template":"qf-density15-selection-forced-initial-order/[23,2,\"506\",\"12144\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=24, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(23, 2, 506, 12144, 1/24)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (23, 2, 506, 12144, 1/24)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r06-permutation-combination-0100","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"629e153abdbc5ae903a68c56a3eabd3a17c1043cdadad2add51815957ba30311","family_id":"qf-density15-selection-forced-initial-order","seed":202610156100,"semantic_template":"qf-density15-selection-forced-initial-order/[24,2,\"552\",\"13800\"]"},"fields":{"title":"Completions after Fixing an Initial Ordered Prefix","stem":"Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=25, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).","answer":{"kind":"expression","value":"(24, 2, 552, 13800, 1/25)","explanation":"Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (24, 2, 552, 13800, 1/25)."},"skills":["Completions after Fixing an Initial Ordered Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}