{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260930-r06-permutation-exact-count-0001","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b70506823be44adde13915068572f85a08ead8680c64b9b8049ea61aff9143fe","family_id":"qf-permutation-exact-count","seed":202609306001,"semantic_template":"qf-permutation-exact-count/{\"k\":0,\"n\":0,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=0 objects in order, without replacement or repetition, from n=0 distinct objects. The empty selection counts once. Find P(0,0).","answer":{"kind":"short_text","value":"1","explanation":"Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0002","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f49eca63d9fae53c58bbf94a16d2ad4888ff0a9d939ea16be9866f3d5bb1ad2b","family_id":"qf-permutation-exact-count","seed":202609306002,"semantic_template":"qf-permutation-exact-count/{\"k\":1,\"n\":1,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=1 objects in order, without replacement or repetition, from n=1 distinct objects. The empty selection counts once. Find P(1,1).","answer":{"kind":"short_text","value":"1","explanation":"Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0003","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51d4e5e4d4c547134dd03a345840d5e2ffc9b7dde61e59ca67be574bc1ff85a0","family_id":"qf-permutation-exact-count","seed":202609306003,"semantic_template":"qf-permutation-exact-count/{\"k\":0,\"n\":2,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=0 objects in order, without replacement or repetition, from n=2 distinct objects. The empty selection counts once. Find P(2,0).","answer":{"kind":"short_text","value":"1","explanation":"Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0004","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03a55a4faa05778962c03cc104f31946ab66fb64088383b42651c12a5936a484","family_id":"qf-permutation-exact-count","seed":202609306004,"semantic_template":"qf-permutation-exact-count/{\"k\":2,\"n\":2,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=2 objects in order, without replacement or repetition, from n=2 distinct objects. The empty selection counts once. Find P(2,2).","answer":{"kind":"short_text","value":"2","explanation":"Multiply the successive available-choice counts exactly to obtain 2; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0005","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ed8970078b467d5eb92b9453226fd1fff9f4f9b2f2c083d7285a7c3dea60598","family_id":"qf-permutation-exact-count","seed":202609306005,"semantic_template":"qf-permutation-exact-count/{\"k\":1,\"n\":3,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=1 objects in order, without replacement or repetition, from n=3 distinct objects. The empty selection counts once. Find P(3,1).","answer":{"kind":"short_text","value":"3","explanation":"Multiply the successive available-choice counts exactly to obtain 3; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0006","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"653a4f9f334ed28d9e6509f415d502f68ff29bd12da25fd6319522c59946d87b","family_id":"qf-permutation-exact-count","seed":202609306006,"semantic_template":"qf-permutation-exact-count/{\"k\":3,\"n\":3,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=3 objects in order, without replacement or repetition, from n=3 distinct objects. The empty selection counts once. Find P(3,3).","answer":{"kind":"short_text","value":"6","explanation":"Multiply the successive available-choice counts exactly to obtain 6; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0007","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eb98451399182ec233a26cdb4ea7f0bd0b5d8bdee54348c198f3aed0f621417","family_id":"qf-permutation-exact-count","seed":202609306007,"semantic_template":"qf-permutation-exact-count/{\"k\":2,\"n\":4,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=2 objects in order, without replacement or repetition, from n=4 distinct objects. The empty selection counts once. Find P(4,2).","answer":{"kind":"short_text","value":"12","explanation":"Multiply the successive available-choice counts exactly to obtain 12; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0008","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9a056ca1818cef7569d9d94105f759fddf00f1421339ddbf3591c4138bc8ac0","family_id":"qf-permutation-exact-count","seed":202609306008,"semantic_template":"qf-permutation-exact-count/{\"k\":4,\"n\":4,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=4 objects in order, without replacement or repetition, from n=4 distinct objects. The empty selection counts once. Find P(4,4).","answer":{"kind":"short_text","value":"24","explanation":"Multiply the successive available-choice counts exactly to obtain 24; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0009","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3da56514c388673e8af973f18d4c97d714f49928bf08c1e097cf554c4316ba38","family_id":"qf-permutation-exact-count","seed":202609306009,"semantic_template":"qf-permutation-exact-count/{\"k\":0,\"n\":5,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=0 objects in order, without replacement or repetition, from n=5 distinct objects. The empty selection counts once. Find P(5,0).","answer":{"kind":"short_text","value":"1","explanation":"Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0010","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78f631cb2a66593c039b17d367f799128ba0be09cfd4673269c024851f0d9e25","family_id":"qf-permutation-exact-count","seed":202609306010,"semantic_template":"qf-permutation-exact-count/{\"k\":1,\"n\":5,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=1 objects in order, without replacement or repetition, from n=5 distinct objects. The empty selection counts once. Find P(5,1).","answer":{"kind":"short_text","value":"5","explanation":"Multiply the successive available-choice counts exactly to obtain 5; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0011","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88a1450997900889129f2f8655ec4ac3a8ed00be0ab43caab63de1003d6d592e","family_id":"qf-permutation-exact-count","seed":202609306011,"semantic_template":"qf-permutation-exact-count/{\"k\":2,\"n\":5,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=2 objects in order, without replacement or repetition, from n=5 distinct objects. The empty selection counts once. Find P(5,2).","answer":{"kind":"short_text","value":"20","explanation":"Multiply the successive available-choice counts exactly to obtain 20; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0012","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed1acc47b6ce791f5667a11e672aaa82adf385220de256b439dcea1e65448176","family_id":"qf-permutation-exact-count","seed":202609306012,"semantic_template":"qf-permutation-exact-count/{\"k\":5,\"n\":5,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=5 objects in order, without replacement or repetition, from n=5 distinct objects. The empty selection counts once. Find P(5,5).","answer":{"kind":"short_text","value":"120","explanation":"Multiply the successive available-choice counts exactly to obtain 120; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0013","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07a2dd11113c6f06575a332e2afbd47b8b3681f17996a0601d8cb90c1f3ba384","family_id":"qf-permutation-exact-count","seed":202609306013,"semantic_template":"qf-permutation-exact-count/{\"k\":1,\"n\":6,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=1 objects in order, without replacement or repetition, from n=6 distinct objects. The empty selection counts once. Find P(6,1).","answer":{"kind":"short_text","value":"6","explanation":"Multiply the successive available-choice counts exactly to obtain 6; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0014","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b0e7cf748604dbfb1b7fba7b427f1ae6bec0be1b4b98a6fbc592bf1ef8506d5","family_id":"qf-permutation-exact-count","seed":202609306014,"semantic_template":"qf-permutation-exact-count/{\"k\":3,\"n\":6,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=3 objects in order, without replacement or repetition, from n=6 distinct objects. The empty selection counts once. Find P(6,3).","answer":{"kind":"short_text","value":"120","explanation":"Multiply the successive available-choice counts exactly to obtain 120; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0015","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78323ab667273afaa0ae718a350c78325ac3e8dfff6e0b6bbcac14306f952217","family_id":"qf-permutation-exact-count","seed":202609306015,"semantic_template":"qf-permutation-exact-count/{\"k\":6,\"n\":6,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=6 objects in order, without replacement or repetition, from n=6 distinct objects. The empty selection counts once. Find P(6,6).","answer":{"kind":"short_text","value":"720","explanation":"Multiply the successive available-choice counts exactly to obtain 720; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0016","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75088b2745a4efa3e97a9f50ab0002d3596df7c08b03e2e0852d42c312328a84","family_id":"qf-permutation-exact-count","seed":202609306016,"semantic_template":"qf-permutation-exact-count/{\"k\":2,\"n\":7,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=2 objects in order, without replacement or repetition, from n=7 distinct objects. The empty selection counts once. Find P(7,2).","answer":{"kind":"short_text","value":"42","explanation":"Multiply the successive available-choice counts exactly to obtain 42; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0017","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6457395abeea445f8ecb6990af0d02e4a7040d06b3363acac2203ace60f73121","family_id":"qf-permutation-exact-count","seed":202609306017,"semantic_template":"qf-permutation-exact-count/{\"k\":4,\"n\":7,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=4 objects in order, without replacement or repetition, from n=7 distinct objects. The empty selection counts once. Find P(7,4).","answer":{"kind":"short_text","value":"840","explanation":"Multiply the successive available-choice counts exactly to obtain 840; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0018","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5451b00911c7662c1d38f126c2e9f4c200d5909044747c6f300491a042483e21","family_id":"qf-permutation-exact-count","seed":202609306018,"semantic_template":"qf-permutation-exact-count/{\"k\":7,\"n\":7,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=7 objects in order, without replacement or repetition, from n=7 distinct objects. The empty selection counts once. Find P(7,7).","answer":{"kind":"short_text","value":"5040","explanation":"Multiply the successive available-choice counts exactly to obtain 5040; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0019","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"156a5d92e869868b7e84896b3e53a09a34ab085b7eb6d952ae59de94b121201a","family_id":"qf-permutation-exact-count","seed":202609306019,"semantic_template":"qf-permutation-exact-count/{\"k\":0,\"n\":8,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=0 objects in order, without replacement or repetition, from n=8 distinct objects. The empty selection counts once. Find P(8,0).","answer":{"kind":"short_text","value":"1","explanation":"Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0020","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ad366b6065caf1eb99a57cb51349b0b981e29714e1bcb27bf7d670525eb15dc","family_id":"qf-permutation-exact-count","seed":202609306020,"semantic_template":"qf-permutation-exact-count/{\"k\":3,\"n\":8,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=3 objects in order, without replacement or repetition, from n=8 distinct objects. The empty selection counts once. Find P(8,3).","answer":{"kind":"short_text","value":"336","explanation":"Multiply the successive available-choice counts exactly to obtain 336; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0021","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93ba60ceba7f65b8d5ab485b7417fdecbe986bd313a46099c80a1e82bdbe4e40","family_id":"qf-permutation-exact-count","seed":202609306021,"semantic_template":"qf-permutation-exact-count/{\"k\":8,\"n\":8,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=8 objects in order, without replacement or repetition, from n=8 distinct objects. The empty selection counts once. Find P(8,8).","answer":{"kind":"short_text","value":"40320","explanation":"Multiply the successive available-choice counts exactly to obtain 40320; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0022","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec783595901a21e8a66e25e6a207416db672735725092d7e2d784c71fa54a293","family_id":"qf-permutation-exact-count","seed":202609306022,"semantic_template":"qf-permutation-exact-count/{\"k\":1,\"n\":9,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=1 objects in order, without replacement or repetition, from n=9 distinct objects. The empty selection counts once. Find P(9,1).","answer":{"kind":"short_text","value":"9","explanation":"Multiply the successive available-choice counts exactly to obtain 9; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0023","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e2a9309baa4010f558412a8d85782734640b6d30e7334c0a8fb9f041831dfbd","family_id":"qf-permutation-exact-count","seed":202609306023,"semantic_template":"qf-permutation-exact-count/{\"k\":4,\"n\":9,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=4 objects in order, without replacement or repetition, from n=9 distinct objects. The empty selection counts once. Find P(9,4).","answer":{"kind":"short_text","value":"3024","explanation":"Multiply the successive available-choice counts exactly to obtain 3024; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0024","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b15ca8d90735ef36f90662c98f0e96ed1258cf0a4bc4b55c0ad16a2a5d9282c5","family_id":"qf-permutation-exact-count","seed":202609306024,"semantic_template":"qf-permutation-exact-count/{\"k\":5,\"n\":10,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=5 objects in order, without replacement or repetition, from n=10 distinct objects. The empty selection counts once. Find P(10,5).","answer":{"kind":"short_text","value":"30240","explanation":"Multiply the successive available-choice counts exactly to obtain 30240; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-permutation-exact-count-0025","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a57ef36688fbb97a388fc0f4f496df503051d2cf3c09081914fe6d1ab1595657","family_id":"qf-permutation-exact-count","seed":202609306025,"semantic_template":"qf-permutation-exact-count/{\"k\":6,\"n\":12,\"operation\":\"permutation\"}"},"fields":{"title":"Exact Ordered Selection Count","stem":"Select k=6 objects in order, without replacement or repetition, from n=12 distinct objects. The empty selection counts once. Find P(12,6).","answer":{"kind":"short_text","value":"665280","explanation":"Multiply the successive available-choice counts exactly to obtain 665280; when k=0 the empty product is 1."},"skills":["distinct objects","ordered 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-20260930-r06-combination-exact-count-0001","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8610efb84508278534ff50ee9088b898bcd66bf9f99b6eb33ac707927e6f8d43","family_id":"qf-combination-exact-count","seed":202609306101,"semantic_template":"qf-combination-exact-count/{\"k\":0,\"n\":0,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=0 objects without order, replacement, or repetition from n=0 distinct objects. The empty selection counts once. Find C(0,0).","answer":{"kind":"short_text","value":"1","explanation":"Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0002","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66f0c6c49c6a97a9f77e4dac7f1c1e80159ea637cbe6f31e651a3b654374101b","family_id":"qf-combination-exact-count","seed":202609306102,"semantic_template":"qf-combination-exact-count/{\"k\":1,\"n\":1,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=1 objects without order, replacement, or repetition from n=1 distinct objects. The empty selection counts once. Find C(1,1).","answer":{"kind":"short_text","value":"1","explanation":"Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0003","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6bf0b0acd766b053342422bff247f28b4ad7886dd3033161fabbd3a297c5d9d","family_id":"qf-combination-exact-count","seed":202609306103,"semantic_template":"qf-combination-exact-count/{\"k\":0,\"n\":2,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=0 objects without order, replacement, or repetition from n=2 distinct objects. The empty selection counts once. Find C(2,0).","answer":{"kind":"short_text","value":"1","explanation":"Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0004","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78df56b824ce975b03272dbf42f60792d09d14894babd7bc49573059d821a4de","family_id":"qf-combination-exact-count","seed":202609306104,"semantic_template":"qf-combination-exact-count/{\"k\":2,\"n\":2,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=2 objects without order, replacement, or repetition from n=2 distinct objects. The empty selection counts once. Find C(2,2).","answer":{"kind":"short_text","value":"1","explanation":"Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0005","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3023480ac573ad5ecc2c6efc05b4eb67a012b81990cd0c42737b7ea0db5e9af","family_id":"qf-combination-exact-count","seed":202609306105,"semantic_template":"qf-combination-exact-count/{\"k\":1,\"n\":3,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=1 objects without order, replacement, or repetition from n=3 distinct objects. The empty selection counts once. Find C(3,1).","answer":{"kind":"short_text","value":"3","explanation":"Count the k-element subsets exactly to obtain 3; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0006","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b1d7339151acb4800dd8c64ffd741176b4d45bbf58c774431fe7b66f9239120","family_id":"qf-combination-exact-count","seed":202609306106,"semantic_template":"qf-combination-exact-count/{\"k\":3,\"n\":3,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=3 objects without order, replacement, or repetition from n=3 distinct objects. The empty selection counts once. Find C(3,3).","answer":{"kind":"short_text","value":"1","explanation":"Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0007","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"713ea171b03eb9246dc9156a034307449cb80dcc965ab883c2f3cc1c9bb64dd9","family_id":"qf-combination-exact-count","seed":202609306107,"semantic_template":"qf-combination-exact-count/{\"k\":2,\"n\":4,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=2 objects without order, replacement, or repetition from n=4 distinct objects. The empty selection counts once. Find C(4,2).","answer":{"kind":"short_text","value":"6","explanation":"Count the k-element subsets exactly to obtain 6; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0008","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b27281d5a4dea44c7f3feb80e675e9ce6e36f05056dfd943333a091b7430d833","family_id":"qf-combination-exact-count","seed":202609306108,"semantic_template":"qf-combination-exact-count/{\"k\":4,\"n\":4,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=4 objects without order, replacement, or repetition from n=4 distinct objects. The empty selection counts once. Find C(4,4).","answer":{"kind":"short_text","value":"1","explanation":"Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0009","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f155c8d9f128725919991f903fff0ee688476d47629e062c1c5e7e99be4d1cf1","family_id":"qf-combination-exact-count","seed":202609306109,"semantic_template":"qf-combination-exact-count/{\"k\":0,\"n\":5,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=0 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,0).","answer":{"kind":"short_text","value":"1","explanation":"Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0010","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2afd710f65ac601c3aa485931a6e1b708672abb2482c89e546f1beac1dfc922","family_id":"qf-combination-exact-count","seed":202609306110,"semantic_template":"qf-combination-exact-count/{\"k\":1,\"n\":5,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=1 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,1).","answer":{"kind":"short_text","value":"5","explanation":"Count the k-element subsets exactly to obtain 5; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0011","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fab4ae6489843a35d9715dda9903be03bf1ad582bce949b1be8bebd50682c01","family_id":"qf-combination-exact-count","seed":202609306111,"semantic_template":"qf-combination-exact-count/{\"k\":2,\"n\":5,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=2 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,2).","answer":{"kind":"short_text","value":"10","explanation":"Count the k-element subsets exactly to obtain 10; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0012","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00d6b0e17a48de12cbd98adeb6a0f753bee4755c44cc6a0feffb41a263f68d7a","family_id":"qf-combination-exact-count","seed":202609306112,"semantic_template":"qf-combination-exact-count/{\"k\":5,\"n\":5,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=5 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,5).","answer":{"kind":"short_text","value":"1","explanation":"Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0013","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e8b694f4d6ebb080933a3f716b5dcc0feccdb5a8a88e490917138b062bf15fb","family_id":"qf-combination-exact-count","seed":202609306113,"semantic_template":"qf-combination-exact-count/{\"k\":1,\"n\":6,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=1 objects without order, replacement, or repetition from n=6 distinct objects. The empty selection counts once. Find C(6,1).","answer":{"kind":"short_text","value":"6","explanation":"Count the k-element subsets exactly to obtain 6; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0014","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c21bbdf01f6f9ed43025d485df288d53e0a230e8767f63d0f9801b3bcf493a38","family_id":"qf-combination-exact-count","seed":202609306114,"semantic_template":"qf-combination-exact-count/{\"k\":3,\"n\":6,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=3 objects without order, replacement, or repetition from n=6 distinct objects. The empty selection counts once. Find C(6,3).","answer":{"kind":"short_text","value":"20","explanation":"Count the k-element subsets exactly to obtain 20; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0015","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4aa40c05f51e40f18373dd46131579144121f02715d07f7dd5d62cfb8408da02","family_id":"qf-combination-exact-count","seed":202609306115,"semantic_template":"qf-combination-exact-count/{\"k\":6,\"n\":6,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=6 objects without order, replacement, or repetition from n=6 distinct objects. The empty selection counts once. Find C(6,6).","answer":{"kind":"short_text","value":"1","explanation":"Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0016","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c3893bd44c54ec16ff15be8142e8b12846f515609a53434d9325205b091850f","family_id":"qf-combination-exact-count","seed":202609306116,"semantic_template":"qf-combination-exact-count/{\"k\":2,\"n\":7,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=2 objects without order, replacement, or repetition from n=7 distinct objects. The empty selection counts once. Find C(7,2).","answer":{"kind":"short_text","value":"21","explanation":"Count the k-element subsets exactly to obtain 21; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0017","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0ff19c1732dc7f58800c5b1c9abc8a2d67e5f37af45c549ea2a452bdfbd7d03","family_id":"qf-combination-exact-count","seed":202609306117,"semantic_template":"qf-combination-exact-count/{\"k\":4,\"n\":7,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=4 objects without order, replacement, or repetition from n=7 distinct objects. The empty selection counts once. Find C(7,4).","answer":{"kind":"short_text","value":"35","explanation":"Count the k-element subsets exactly to obtain 35; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0018","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c3b9d9e3ef47b4b409f9c6a135c4b2557d8c790698138e85343e95422cb6ee8","family_id":"qf-combination-exact-count","seed":202609306118,"semantic_template":"qf-combination-exact-count/{\"k\":7,\"n\":7,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=7 objects without order, replacement, or repetition from n=7 distinct objects. The empty selection counts once. Find C(7,7).","answer":{"kind":"short_text","value":"1","explanation":"Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0019","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71e2bd0b54e9794b2095c675c686e969fb50382addb494567cddee9ad15a57e8","family_id":"qf-combination-exact-count","seed":202609306119,"semantic_template":"qf-combination-exact-count/{\"k\":0,\"n\":8,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=0 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,0).","answer":{"kind":"short_text","value":"1","explanation":"Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0020","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e780797cbc947d3fcd6eb8a6cbc47ef30549171b2b046fc5a8ec79a70b548208","family_id":"qf-combination-exact-count","seed":202609306120,"semantic_template":"qf-combination-exact-count/{\"k\":3,\"n\":8,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=3 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,3).","answer":{"kind":"short_text","value":"56","explanation":"Count the k-element subsets exactly to obtain 56; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0021","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b24ae8869a1c575b1c735b1473eaf5b27e1c63a139c9accf421c9d6b99f235d","family_id":"qf-combination-exact-count","seed":202609306121,"semantic_template":"qf-combination-exact-count/{\"k\":8,\"n\":8,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=8 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,8).","answer":{"kind":"short_text","value":"1","explanation":"Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0022","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6756230ae3c3d1ad47bf6098768810d76bc54262eaf72b8cea8c9191aa5ef2ee","family_id":"qf-combination-exact-count","seed":202609306122,"semantic_template":"qf-combination-exact-count/{\"k\":1,\"n\":9,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=1 objects without order, replacement, or repetition from n=9 distinct objects. The empty selection counts once. Find C(9,1).","answer":{"kind":"short_text","value":"9","explanation":"Count the k-element subsets exactly to obtain 9; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0023","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcb81f01d828f570add7f7d7b92d21253d5b3219ecc1377459e43d025525fbb5","family_id":"qf-combination-exact-count","seed":202609306123,"semantic_template":"qf-combination-exact-count/{\"k\":4,\"n\":9,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=4 objects without order, replacement, or repetition from n=9 distinct objects. The empty selection counts once. Find C(9,4).","answer":{"kind":"short_text","value":"126","explanation":"Count the k-element subsets exactly to obtain 126; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0024","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6a7ea594a31c0aacb4469ce23b4bdaa74c4a89660720431bfecc793617f8af8","family_id":"qf-combination-exact-count","seed":202609306124,"semantic_template":"qf-combination-exact-count/{\"k\":5,\"n\":10,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=5 objects without order, replacement, or repetition from n=10 distinct objects. The empty selection counts once. Find C(10,5).","answer":{"kind":"short_text","value":"252","explanation":"Count the k-element subsets exactly to obtain 252; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-combination-exact-count-0025","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"884d61ddb435444adef96c08cd6ac567b1e76fcb62cab093385226faff628c1c","family_id":"qf-combination-exact-count","seed":202609306125,"semantic_template":"qf-combination-exact-count/{\"k\":6,\"n\":12,\"operation\":\"combination\"}"},"fields":{"title":"Exact Unordered Selection Count","stem":"Select k=6 objects without order, replacement, or repetition from n=12 distinct objects. The empty selection counts once. Find C(12,6).","answer":{"kind":"short_text","value":"924","explanation":"Count the k-element subsets exactly to obtain 924; the empty subset is one valid selection."},"skills":["distinct objects","unordered 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-20260930-r06-selection-model-choice-0001","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47471cd21e27619faa00cf9ccbb22f972e50310a744b6e27f64f70fcca626d90","family_id":"qf-selection-model-choice","seed":202609306201,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":0,\"n\":0,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=0 objects with order distinguished, without replacement or repetition, from n=0 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0002","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14d6210e0434ad4844ae7ae686563a86f63eaec94257e085fda5d58cc7df9b5b","family_id":"qf-selection-model-choice","seed":202609306202,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":1,\"n\":1,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=1 objects with order ignored, without replacement or repetition, from n=1 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"combination","explanation":"The objects are distinct and selection is without replacement. Because order is ignored, use the combination model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0003","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"703d48923f86c1040a1c53a64720f5ad1e76389d1ebbd628b344335e0667e7a3","family_id":"qf-selection-model-choice","seed":202609306203,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":0,\"n\":2,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=0 objects with order distinguished, without replacement or repetition, from n=2 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0004","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ccba526047262e8536607f8ce6367c3cc19b1454ebd1d617487b4cda4bf1a24","family_id":"qf-selection-model-choice","seed":202609306204,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":2,\"n\":2,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=2 objects with order ignored, without replacement or repetition, from n=2 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"combination","explanation":"The objects are distinct and selection is without replacement. Because order is ignored, use the combination model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0005","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cad423702ced168d558a20facb9845c5cec55ea7d9690167b38ee96c22389485","family_id":"qf-selection-model-choice","seed":202609306205,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":1,\"n\":3,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=1 objects with order distinguished, without replacement or repetition, from n=3 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0006","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44965a67ef2f71c879ba4eed5ab57469275369417f121ecfe474602a0d636cda","family_id":"qf-selection-model-choice","seed":202609306206,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":3,\"n\":3,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=3 objects with order ignored, without replacement or repetition, from n=3 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"combination","explanation":"The objects are distinct and selection is without replacement. Because order is ignored, use the combination model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0007","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98c5b9a08bfdfd68dac39cc46656adf0d500199c722f1ab9d4b0c898350a72d0","family_id":"qf-selection-model-choice","seed":202609306207,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":2,\"n\":4,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=2 objects with order distinguished, without replacement or repetition, from n=4 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0008","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc40d89abf5f949362d83e4c8047f411f42456cb518a3741391fa8f9765919da","family_id":"qf-selection-model-choice","seed":202609306208,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":4,\"n\":4,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=4 objects with order ignored, without replacement or repetition, from n=4 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"combination","explanation":"The objects are distinct and selection is without replacement. Because order is ignored, use the combination model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0009","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bbd9382ffac5db66b97fa7065b6f14fde9b26de4047860b2d35529a90bcd265","family_id":"qf-selection-model-choice","seed":202609306209,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":0,\"n\":5,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=0 objects with order distinguished, without replacement or repetition, from n=5 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0010","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04aba07d3c2d87f3248aa278cdf89abb271dbb348e6c5aaac72130a2c1816b72","family_id":"qf-selection-model-choice","seed":202609306210,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":1,\"n\":5,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=1 objects with order ignored, without replacement or repetition, from n=5 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"combination","explanation":"The objects are distinct and selection is without replacement. Because order is ignored, use the combination model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0011","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd85ca20b53a6881975255f8cfee3629556dafbf681991dc0322e7c00d9e172d","family_id":"qf-selection-model-choice","seed":202609306211,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":2,\"n\":5,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=2 objects with order distinguished, without replacement or repetition, from n=5 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0012","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4285ea2ec23f2bd4cf5dba62a812b5bdff868339f2da81f639327adb29973641","family_id":"qf-selection-model-choice","seed":202609306212,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":5,\"n\":5,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=5 objects with order ignored, without replacement or repetition, from n=5 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"combination","explanation":"The objects are distinct and selection is without replacement. Because order is ignored, use the combination model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0013","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"745027c6e27f60690db535cae03e8b014f36d17d242a326f6b6d98bd94c9e277","family_id":"qf-selection-model-choice","seed":202609306213,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":1,\"n\":6,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=1 objects with order distinguished, without replacement or repetition, from n=6 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0014","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccd8540c6d73b3f113ae70803256580c696641442e59ea3be5b547e8d8da7fd5","family_id":"qf-selection-model-choice","seed":202609306214,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":3,\"n\":6,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=3 objects with order ignored, without replacement or repetition, from n=6 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"combination","explanation":"The objects are distinct and selection is without replacement. Because order is ignored, use the combination model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0015","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ac066b2cec37a381219929278fb0edc8077d42f29e7789afd593cc8c9281dcf","family_id":"qf-selection-model-choice","seed":202609306215,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":6,\"n\":6,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=6 objects with order distinguished, without replacement or repetition, from n=6 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0016","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a49a18e90c3bb7ec720079809d3d98c00dae99b4cd24f87bd4455748220fdebd","family_id":"qf-selection-model-choice","seed":202609306216,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":2,\"n\":7,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=2 objects with order ignored, without replacement or repetition, from n=7 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"combination","explanation":"The objects are distinct and selection is without replacement. Because order is ignored, use the combination model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0017","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a78338dbf095f02442e569636bade1e4210577ed757601d35cecce495e3c554","family_id":"qf-selection-model-choice","seed":202609306217,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":4,\"n\":7,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=4 objects with order distinguished, without replacement or repetition, from n=7 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0018","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b4715c98d20131b6e780d9fd0f219c03160abb69c0eea31a50108f91039319f","family_id":"qf-selection-model-choice","seed":202609306218,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":7,\"n\":7,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=7 objects with order ignored, without replacement or repetition, from n=7 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"combination","explanation":"The objects are distinct and selection is without replacement. Because order is ignored, use the combination model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0019","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dba79217aacf9e4b7c714a2acf98621c468560c1570eadeb2c27ad45aa75813","family_id":"qf-selection-model-choice","seed":202609306219,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":0,\"n\":8,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=0 objects with order distinguished, without replacement or repetition, from n=8 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0020","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e518aff6753a5282651a7888d08f4cc9a5025be80eb1a58a80c68d9b6b2db46","family_id":"qf-selection-model-choice","seed":202609306220,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":3,\"n\":8,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=3 objects with order ignored, without replacement or repetition, from n=8 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"combination","explanation":"The objects are distinct and selection is without replacement. Because order is ignored, use the combination model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0021","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"038c8a33ec710314362cb5c20a2730df18f69cd81fc483468cbb3451a45be803","family_id":"qf-selection-model-choice","seed":202609306221,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":8,\"n\":8,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=8 objects with order distinguished, without replacement or repetition, from n=8 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0022","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f4f20a66d8417e539bdf0b45468c77cb65537633f77382c52dbb4de9b5c4bdf","family_id":"qf-selection-model-choice","seed":202609306222,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":1,\"n\":9,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=1 objects with order ignored, without replacement or repetition, from n=9 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"combination","explanation":"The objects are distinct and selection is without replacement. Because order is ignored, use the combination model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0023","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99cc7811a8853b683e1c6addc0c2f4e7761f43d18486c4599ff294d7130f1dfb","family_id":"qf-selection-model-choice","seed":202609306223,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":4,\"n\":9,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=4 objects with order distinguished, without replacement or repetition, from n=9 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0024","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a1b36d0c36596a39309c2ddbf47251d80293d0ad428980d025bead4dd63e3f8","family_id":"qf-selection-model-choice","seed":202609306224,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":5,\"n\":10,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=5 objects with order ignored, without replacement or repetition, from n=10 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"combination","explanation":"The objects are distinct and selection is without replacement. Because order is ignored, use the combination model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-selection-model-choice-0025","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"807c1367411c5c0624387ea570103590b8da5b18a6ec32dd7f865553377de775","family_id":"qf-selection-model-choice","seed":202609306225,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":6,\"n\":12,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"},"fields":{"title":"Choosing a Selection Model","stem":"Select k=6 objects with order distinguished, without replacement or repetition, from n=12 distinct objects. Should the model be a permutation or a combination?","answer":{"kind":"short_text","value":"permutation","explanation":"The objects are distinct and selection is without replacement. Because order is significant, use the permutation model."},"skills":["ordering condition","without replacement"],"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-20260930-r06-permutation-combination-quotient-0001","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ac4ec4844a1fa74ec94f2ee1ca6464b429eb161cd7fbc6a4e87eb154d95ea77","family_id":"qf-permutation-combination-quotient","seed":202609306301,"semantic_template":"qf-permutation-combination-quotient/{\"k\":0,\"n\":0,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=0 distinct objects without replacement from n=0, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"1","explanation":"Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0002","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7963f20362dc48b42afd43e0fb82be17efbbfb48814b476e2a27776419f08ab1","family_id":"qf-permutation-combination-quotient","seed":202609306302,"semantic_template":"qf-permutation-combination-quotient/{\"k\":1,\"n\":1,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=1 distinct objects without replacement from n=1, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"1","explanation":"Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0003","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f7c6144c59ce6659a786930fbb722095bc603a207a4369f5d1b9707502b9d33","family_id":"qf-permutation-combination-quotient","seed":202609306303,"semantic_template":"qf-permutation-combination-quotient/{\"k\":0,\"n\":2,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=0 distinct objects without replacement from n=2, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"1","explanation":"Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0004","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0bf13c5df54ab2bc73297d368b1df584f7e2cdd550af5d7d48f37a9055d4b7e","family_id":"qf-permutation-combination-quotient","seed":202609306304,"semantic_template":"qf-permutation-combination-quotient/{\"k\":2,\"n\":2,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=2 distinct objects without replacement from n=2, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"2","explanation":"Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0005","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d580f298f4df639396a0c0333091f42ab28eaea679cc585dd737d78431ed469","family_id":"qf-permutation-combination-quotient","seed":202609306305,"semantic_template":"qf-permutation-combination-quotient/{\"k\":1,\"n\":3,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=1 distinct objects without replacement from n=3, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"1","explanation":"Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0006","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cd559219801d050c21ba2990687ae37dd9b3f8071b931db1e445cd22bd5f234","family_id":"qf-permutation-combination-quotient","seed":202609306306,"semantic_template":"qf-permutation-combination-quotient/{\"k\":3,\"n\":3,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=3 distinct objects without replacement from n=3, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"6","explanation":"Each k-element subset has k! orderings, so the quotient is 6; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0007","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3de3b845d21257b1154e65d6906b03580db6c65333a8693f4a6736f69ee8aee5","family_id":"qf-permutation-combination-quotient","seed":202609306307,"semantic_template":"qf-permutation-combination-quotient/{\"k\":2,\"n\":4,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=2 distinct objects without replacement from n=4, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"2","explanation":"Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0008","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0535ef15237218f4b69a1056b00be894998e9cf668b2a6f70675e73483ca413c","family_id":"qf-permutation-combination-quotient","seed":202609306308,"semantic_template":"qf-permutation-combination-quotient/{\"k\":4,\"n\":4,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=4 distinct objects without replacement from n=4, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"24","explanation":"Each k-element subset has k! orderings, so the quotient is 24; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0009","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f3cd7afff28c98b8873ddc8d89cc443e5daa90bf23ddff65ee248b5fb76561d","family_id":"qf-permutation-combination-quotient","seed":202609306309,"semantic_template":"qf-permutation-combination-quotient/{\"k\":0,\"n\":5,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=0 distinct objects without replacement from n=5, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"1","explanation":"Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0010","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"915775ebcd8b25d316cde853d4db558ef054a41196e3e5ca96755ea779e74716","family_id":"qf-permutation-combination-quotient","seed":202609306310,"semantic_template":"qf-permutation-combination-quotient/{\"k\":1,\"n\":5,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=1 distinct objects without replacement from n=5, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"1","explanation":"Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0011","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d36f2be0b0e90c157322b3bf6708ef77d562cae396c78b9d39d36526a89468be","family_id":"qf-permutation-combination-quotient","seed":202609306311,"semantic_template":"qf-permutation-combination-quotient/{\"k\":2,\"n\":5,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=2 distinct objects without replacement from n=5, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"2","explanation":"Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0012","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67faef5e9bb154e79ef0b47172f67dbba0036531d98ed113d97ea8d6a91cf609","family_id":"qf-permutation-combination-quotient","seed":202609306312,"semantic_template":"qf-permutation-combination-quotient/{\"k\":5,\"n\":5,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=5 distinct objects without replacement from n=5, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"120","explanation":"Each k-element subset has k! orderings, so the quotient is 120; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0013","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57542b62dc8e415d06652efba46dbe3fbed76ad63ad13dc523bc9ddea7df43e1","family_id":"qf-permutation-combination-quotient","seed":202609306313,"semantic_template":"qf-permutation-combination-quotient/{\"k\":1,\"n\":6,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=1 distinct objects without replacement from n=6, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"1","explanation":"Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0014","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5ec25b5ad2ce995b6de1fbc6e7b7781fa388bc37fa1a3c533ad0cbfb105be69","family_id":"qf-permutation-combination-quotient","seed":202609306314,"semantic_template":"qf-permutation-combination-quotient/{\"k\":3,\"n\":6,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=3 distinct objects without replacement from n=6, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"6","explanation":"Each k-element subset has k! orderings, so the quotient is 6; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0015","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c67f0f31322db53581dd78d192bf48737c5bdbd452ed5db4fcd670942d833223","family_id":"qf-permutation-combination-quotient","seed":202609306315,"semantic_template":"qf-permutation-combination-quotient/{\"k\":6,\"n\":6,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=6 distinct objects without replacement from n=6, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"720","explanation":"Each k-element subset has k! orderings, so the quotient is 720; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0016","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fadc06f39e07abc019388c07b0802902257c4459a172ec7c211a7135e91c5ce","family_id":"qf-permutation-combination-quotient","seed":202609306316,"semantic_template":"qf-permutation-combination-quotient/{\"k\":2,\"n\":7,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=2 distinct objects without replacement from n=7, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"2","explanation":"Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0017","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99b69a4812e315e80c63b0971ed278b5b410c011f1ec037e8469e557cf3c28ad","family_id":"qf-permutation-combination-quotient","seed":202609306317,"semantic_template":"qf-permutation-combination-quotient/{\"k\":4,\"n\":7,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=4 distinct objects without replacement from n=7, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"24","explanation":"Each k-element subset has k! orderings, so the quotient is 24; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0018","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6746f5a5fd592ecaca7a3f19b9ab187368fea9aecfe75e9c95be50a8abbd6962","family_id":"qf-permutation-combination-quotient","seed":202609306318,"semantic_template":"qf-permutation-combination-quotient/{\"k\":7,\"n\":7,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=7 distinct objects without replacement from n=7, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"5040","explanation":"Each k-element subset has k! orderings, so the quotient is 5040; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0019","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"830497bd8a57d602d16dabd9194a72185aca1d02f8d86a65bb24d011d888fc0d","family_id":"qf-permutation-combination-quotient","seed":202609306319,"semantic_template":"qf-permutation-combination-quotient/{\"k\":0,\"n\":8,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=0 distinct objects without replacement from n=8, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"1","explanation":"Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0020","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f81090d910fa2d4f0d83ec442666ba472a1a3e7776698b0dd2102e6d640fdf6","family_id":"qf-permutation-combination-quotient","seed":202609306320,"semantic_template":"qf-permutation-combination-quotient/{\"k\":3,\"n\":8,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=3 distinct objects without replacement from n=8, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"6","explanation":"Each k-element subset has k! orderings, so the quotient is 6; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0021","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd861db73f07a757e9aee9c1393bd58d54d4fc87538e4064e22a845d4514e91c","family_id":"qf-permutation-combination-quotient","seed":202609306321,"semantic_template":"qf-permutation-combination-quotient/{\"k\":8,\"n\":8,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=8 distinct objects without replacement from n=8, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"40320","explanation":"Each k-element subset has k! orderings, so the quotient is 40320; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0022","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73264d75d5be15d163f1e83d369dc3b8f873ae4c9ca450f1cd1bb763bbe8f595","family_id":"qf-permutation-combination-quotient","seed":202609306322,"semantic_template":"qf-permutation-combination-quotient/{\"k\":1,\"n\":9,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=1 distinct objects without replacement from n=9, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"1","explanation":"Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0023","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fc863af0d23822d4126e6c17db6cff9089f201137ec644b1042d07143caf188","family_id":"qf-permutation-combination-quotient","seed":202609306323,"semantic_template":"qf-permutation-combination-quotient/{\"k\":4,\"n\":9,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=4 distinct objects without replacement from n=9, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"24","explanation":"Each k-element subset has k! orderings, so the quotient is 24; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0024","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6f84d31d2961b39cc660ca74092eafb89e9ce07315d570cdff1560a678092f6","family_id":"qf-permutation-combination-quotient","seed":202609306324,"semantic_template":"qf-permutation-combination-quotient/{\"k\":5,\"n\":10,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=5 distinct objects without replacement from n=10, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"120","explanation":"Each k-element subset has k! orderings, so the quotient is 120; the empty subset also has one ordering."},"skills":["orderings of each subset","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-20260930-r06-permutation-combination-quotient-0025","batch_id":"batch-20260930-r06-permutation-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7e311700d39c89b618ce0e9db66c44238ed10ace44e48bc2e4f962f5ed33a77","family_id":"qf-permutation-combination-quotient","seed":202609306325,"semantic_template":"qf-permutation-combination-quotient/{\"k\":6,\"n\":12,\"operation\":\"quotient\"}"},"fields":{"title":"Quotient of Permutation and Combination Counts","stem":"For selection of k=6 distinct objects without replacement from n=12, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.","answer":{"kind":"short_text","value":"720","explanation":"Each k-element subset has k! orderings, so the quotient is 720; the empty subset also has one ordering."},"skills":["orderings of each subset","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."}}