{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r05-binomial-theorem-0001","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c422097c50b54a8aac60229724ba241c40914d2832005c6affff8104d6d110c","family_id":"qf-mod40-binomial-coefficient","seed":202611095001,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":0,\"n\":0}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=0 with 0≤n≤50 and integer k=0 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(0, 0, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (0, 0, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0002","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1079f0edfa70e4cd8e2f0fddafd285ff4a59036b03ecc9f82665265f3c2a161f","family_id":"qf-mod40-binomial-coefficient","seed":202611095002,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":1,\"n\":1}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=1 with 0≤n≤50 and integer k=1 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(1, 1, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (1, 1, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0003","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2951afbeeed558db6e06516be4744a341217b8852b07484472fb94971d5de0f4","family_id":"qf-mod40-binomial-coefficient","seed":202611095003,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":2,\"n\":2}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=2 with 0≤n≤50 and integer k=2 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(2, 2, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (2, 2, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0004","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da5c0da072c76704b05275baee66af904cccea5f925439b85cbe66ed0cf95e51","family_id":"qf-mod40-binomial-coefficient","seed":202611095004,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":3,\"n\":3}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=3 with 0≤n≤50 and integer k=3 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(3, 3, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (3, 3, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0005","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8f5b9d4b46c96cb0a38a9b88f067241508335bfb504f18d12ffe7fefa6d7d5b","family_id":"qf-mod40-binomial-coefficient","seed":202611095005,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":4,\"n\":4}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=4 with 0≤n≤50 and integer k=4 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(4, 4, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (4, 4, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0006","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c24e12a4dcbec40c245eadeb24b486c0f40209bb2ee42fde8ac65490f358a13e","family_id":"qf-mod40-binomial-coefficient","seed":202611095006,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":5,\"n\":5}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=5 with 0≤n≤50 and integer k=5 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(5, 5, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (5, 5, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0007","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57516a5baac20b64c71536209bd84481b5f1e96ca6a28d7422b7a2287b2f6e7e","family_id":"qf-mod40-binomial-coefficient","seed":202611095007,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":6,\"n\":6}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=6 with 0≤n≤50 and integer k=6 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(6, 6, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (6, 6, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0008","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7be81bf022f8740c19abf141683ae22c37a35abffa9dbe3ff219aaf5556b0807","family_id":"qf-mod40-binomial-coefficient","seed":202611095008,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":7,\"n\":7}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=7 with 0≤n≤50 and integer k=7 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(7, 7, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (7, 7, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0009","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c690539069ab95b01b12b18750e37879d3414edff423f85051cad3e2281c6646","family_id":"qf-mod40-binomial-coefficient","seed":202611095009,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":8,\"n\":8}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=8 with 0≤n≤50 and integer k=8 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(8, 8, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (8, 8, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0010","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26bff500dbd6b8c7f69bd45a0d6646ae4e6f4cc4d65d4c1f770120160480f547","family_id":"qf-mod40-binomial-coefficient","seed":202611095010,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":9,\"n\":9}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=9 with 0≤n≤50 and integer k=9 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(9, 9, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (9, 9, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0011","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2054cfd71b41212eb266699acecb9697bcbaf4423e4e4827e7a42eeb9e73a824","family_id":"qf-mod40-binomial-coefficient","seed":202611095011,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":10,\"n\":10}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=10 with 0≤n≤50 and integer k=10 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(10, 10, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (10, 10, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0012","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86aaf4ff21ced8f7828b35ecadd62e1e76933869b57040d0180b76b111bc5b46","family_id":"qf-mod40-binomial-coefficient","seed":202611095012,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":11,\"n\":11}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=11 with 0≤n≤50 and integer k=11 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(11, 11, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (11, 11, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0013","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51e2c4492d6f2acc7850ba21224b65152b02c1ff9e1d999334532df33ece3704","family_id":"qf-mod40-binomial-coefficient","seed":202611095013,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":12,\"n\":12}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=12 with 0≤n≤50 and integer k=12 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(12, 12, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (12, 12, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0014","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47de5da6327b1a0f7b8c87ff6b86abb6db8f6688ed434faae09ca9a4fa01f1a3","family_id":"qf-mod40-binomial-coefficient","seed":202611095014,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":13,\"n\":13}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=13 with 0≤n≤50 and integer k=13 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(13, 13, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (13, 13, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0015","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"550b358758a9130a177e1758dfb31b5e062f38385ad7b0fafced26ab2ed37b45","family_id":"qf-mod40-binomial-coefficient","seed":202611095015,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":14,\"n\":14}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=14 with 0≤n≤50 and integer k=14 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(14, 14, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (14, 14, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0016","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae7705a1f5b251ede693fb557ec309c14e193efba3e304b05098fba68a877352","family_id":"qf-mod40-binomial-coefficient","seed":202611095016,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":15,\"n\":15}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=15 with 0≤n≤50 and integer k=15 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(15, 15, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (15, 15, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0017","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da4f1ccb492af2a591a27e00392f836e9a150b8146f68b09db80c5cbe0925c24","family_id":"qf-mod40-binomial-coefficient","seed":202611095017,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":16,\"n\":16}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=16 with 0≤n≤50 and integer k=16 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(16, 16, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (16, 16, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0018","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9dcbff8632dcec68b29bb44e5e5a098a138fa7e4a2760d00c7f441be45e8a76","family_id":"qf-mod40-binomial-coefficient","seed":202611095018,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":17,\"n\":17}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=17 with 0≤n≤50 and integer k=17 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(17, 17, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (17, 17, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0019","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"945595a250affc40d2a5f50c7108d43f9847784a1735cdfce15fa1a004075da3","family_id":"qf-mod40-binomial-coefficient","seed":202611095019,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":18,\"n\":18}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=18 with 0≤n≤50 and integer k=18 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(18, 18, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (18, 18, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0020","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"527d5380df110b28525946d0cffe37124fb07dc4ef581346cdf6e3d8753a1974","family_id":"qf-mod40-binomial-coefficient","seed":202611095020,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":19,\"n\":19}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=19 with 0≤n≤50 and integer k=19 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(19, 19, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (19, 19, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0021","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34606b508cb09a2fde4e0597cc715ba866919f3f668f72c6ad223852fc52c7dc","family_id":"qf-mod40-binomial-coefficient","seed":202611095021,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":20,\"n\":20}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=20 with 0≤n≤50 and integer k=20 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(20, 20, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (20, 20, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0022","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e185c7d568747b47afcfba5238f75bea5680c21acd6cce1e2bf6c634b30a3d21","family_id":"qf-mod40-binomial-coefficient","seed":202611095022,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":21,\"n\":21}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=21 with 0≤n≤50 and integer k=21 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(21, 21, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (21, 21, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0023","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d751e49ab0a37dfa90a42c0c8824bf32262e1434b4121599fcd7695f37dc4c3b","family_id":"qf-mod40-binomial-coefficient","seed":202611095023,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":22,\"n\":22}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=22 with 0≤n≤50 and integer k=22 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(22, 22, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (22, 22, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0024","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"899dc3cac67e88b7c0b1183f7c2d45b7ac28eddbd7d8509a1d7577e876e4a045","family_id":"qf-mod40-binomial-coefficient","seed":202611095024,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":23,\"n\":23}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=23 with 0≤n≤50 and integer k=23 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(23, 23, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (23, 23, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0025","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c3dc47299a7d7ecdd878bfffc2910f581f965097cc05ad3b1844e73680d157a","family_id":"qf-mod40-binomial-coefficient","seed":202611095025,"semantic_template":"qf-mod40-binomial-coefficient/{\"k\":24,\"n\":24}"},"fields":{"title":"Exact Binomial Coefficients Keep k and n Minus k Distinct","stem":"Use integer n=24 with 0≤n≤50 and integer k=24 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).","answer":{"kind":"expression","value":"(24, 24, 1, 1, Yes)","explanation":"Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (24, 24, 1, 1, Yes)."},"skills":["Exact Binomial Coefficients Keep k and n Minus k Distinct"],"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-20261014-advance-20261109-r05-binomial-theorem-0026","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ab6f1fad6c6b2731837e8c6140d5184db5ed2bb90ec471cbad328ce1196a4d5","family_id":"qf-mod40-binomial-expansion","seed":202611095026,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":0}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=0, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(0, [[0, 1, 0, 0]], constant_one_boundary)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (0, [[0, 1, 0, 0]], constant_one_boundary)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0027","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"789a684f060fec7a772d8a3676aaf0f4ce0c8674afd593a37494588938717da6","family_id":"qf-mod40-binomial-expansion","seed":202611095027,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":1}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=1, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(1, [[0, 1, 1, 0], [1, 1, 0, 1]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (1, [[0, 1, 1, 0], [1, 1, 0, 1]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0028","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c00566b36326b791c4ab9313a94694fafa92b2e45d3b78f0935fd7f56275ac21","family_id":"qf-mod40-binomial-expansion","seed":202611095028,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":2}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=2, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(2, [[0, 1, 2, 0], [1, 2, 1, 1], [2, 1, 0, 2]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (2, [[0, 1, 2, 0], [1, 2, 1, 1], [2, 1, 0, 2]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0029","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf4e1f7f8d78ed60f943b2d479df01c4f91916a75a398eaa11210089550988c2","family_id":"qf-mod40-binomial-expansion","seed":202611095029,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":3}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=3, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(3, [[0, 1, 3, 0], [1, 3, 2, 1], [2, 3, 1, 2], [3, 1, 0, 3]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (3, [[0, 1, 3, 0], [1, 3, 2, 1], [2, 3, 1, 2], [3, 1, 0, 3]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0030","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3aab3830ab5e3dd310ade7aaa3decdad12221ba23cfbe5c57986a4dc6d814082","family_id":"qf-mod40-binomial-expansion","seed":202611095030,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":4}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=4, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(4, [[0, 1, 4, 0], [1, 4, 3, 1], [2, 6, 2, 2], [3, 4, 1, 3], [4, 1, 0, 4]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (4, [[0, 1, 4, 0], [1, 4, 3, 1], [2, 6, 2, 2], [3, 4, 1, 3], [4, 1, 0, 4]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0031","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3c3e4ccc308bcd901b24542db12b5c2a10f1d605f9acf9a7cc4b678b82b2f78","family_id":"qf-mod40-binomial-expansion","seed":202611095031,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":5}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=5, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(5, [[0, 1, 5, 0], [1, 5, 4, 1], [2, 10, 3, 2], [3, 10, 2, 3], [4, 5, 1, 4], [5, 1, 0, 5]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (5, [[0, 1, 5, 0], [1, 5, 4, 1], [2, 10, 3, 2], [3, 10, 2, 3], [4, 5, 1, 4], [5, 1, 0, 5]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0032","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16bf39eb4083814e2793cbfe26a705eae841583efe8149c59412e44ed9c19385","family_id":"qf-mod40-binomial-expansion","seed":202611095032,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":6}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=6, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(6, [[0, 1, 6, 0], [1, 6, 5, 1], [2, 15, 4, 2], [3, 20, 3, 3], [4, 15, 2, 4], [5, 6, 1, 5], [6, 1, 0, 6]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (6, [[0, 1, 6, 0], [1, 6, 5, 1], [2, 15, 4, 2], [3, 20, 3, 3], [4, 15, 2, 4], [5, 6, 1, 5], [6, 1, 0, 6]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0033","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f438c1c6546cfeb2051dba9f214004bdc8d20956d93d4bcf689a8c5b1088d7b5","family_id":"qf-mod40-binomial-expansion","seed":202611095033,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":7}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=7, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(7, [[0, 1, 7, 0], [1, 7, 6, 1], [2, 21, 5, 2], [3, 35, 4, 3], [4, 35, 3, 4], [5, 21, 2, 5], [6, 7, 1, 6], [7, 1, 0, 7]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (7, [[0, 1, 7, 0], [1, 7, 6, 1], [2, 21, 5, 2], [3, 35, 4, 3], [4, 35, 3, 4], [5, 21, 2, 5], [6, 7, 1, 6], [7, 1, 0, 7]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0034","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc9f23a697f7335087267e53580835ab8bfeb8a2899be3d531f640fdfc845f21","family_id":"qf-mod40-binomial-expansion","seed":202611095034,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":8}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=8, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(8, [[0, 1, 8, 0], [1, 8, 7, 1], [2, 28, 6, 2], [3, 56, 5, 3], [4, 70, 4, 4], [5, 56, 3, 5], [6, 28, 2, 6], [7, 8, 1, 7], [8, 1, 0, 8]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (8, [[0, 1, 8, 0], [1, 8, 7, 1], [2, 28, 6, 2], [3, 56, 5, 3], [4, 70, 4, 4], [5, 56, 3, 5], [6, 28, 2, 6], [7, 8, 1, 7], [8, 1, 0, 8]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0035","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9105b70c0fe521b1d2a2e605050446312f8d4bb127fe1cf3e146d9de08b0676","family_id":"qf-mod40-binomial-expansion","seed":202611095035,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":9}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=9, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(9, [[0, 1, 9, 0], [1, 9, 8, 1], [2, 36, 7, 2], [3, 84, 6, 3], [4, 126, 5, 4], [5, 126, 4, 5], [6, 84, 3, 6], [7, 36, 2, 7], [8, 9, 1, 8], [9, 1, 0, 9]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (9, [[0, 1, 9, 0], [1, 9, 8, 1], [2, 36, 7, 2], [3, 84, 6, 3], [4, 126, 5, 4], [5, 126, 4, 5], [6, 84, 3, 6], [7, 36, 2, 7], [8, 9, 1, 8], [9, 1, 0, 9]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0036","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9135df7c9fe9f03626f516048438cdecbfd4bf64e92f7fadcc77178f03abb65b","family_id":"qf-mod40-binomial-expansion","seed":202611095036,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":10}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=10, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(10, [[0, 1, 10, 0], [1, 10, 9, 1], [2, 45, 8, 2], [3, 120, 7, 3], [4, 210, 6, 4], [5, 252, 5, 5], [6, 210, 4, 6], [7, 120, 3, 7], [8, 45, 2, 8], [9, 10, 1, 9], [10, 1, 0, 10]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (10, [[0, 1, 10, 0], [1, 10, 9, 1], [2, 45, 8, 2], [3, 120, 7, 3], [4, 210, 6, 4], [5, 252, 5, 5], [6, 210, 4, 6], [7, 120, 3, 7], [8, 45, 2, 8], [9, 10, 1, 9], [10, 1, 0, 10]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0037","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06cea563d841c3a03e9dc63e8b016edc57b7739a8263f3444d07a3ad6be09fcb","family_id":"qf-mod40-binomial-expansion","seed":202611095037,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":11}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=11, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(11, [[0, 1, 11, 0], [1, 11, 10, 1], [2, 55, 9, 2], [3, 165, 8, 3], [4, 330, 7, 4], [5, 462, 6, 5], [6, 462, 5, 6], [7, 330, 4, 7], [8, 165, 3, 8], [9, 55, 2, 9], [10, 11, 1, 10], [11, 1, 0, 11]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (11, [[0, 1, 11, 0], [1, 11, 10, 1], [2, 55, 9, 2], [3, 165, 8, 3], [4, 330, 7, 4], [5, 462, 6, 5], [6, 462, 5, 6], [7, 330, 4, 7], [8, 165, 3, 8], [9, 55, 2, 9], [10, 11, 1, 10], [11, 1, 0, 11]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0038","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93d2e4a077e414a3942bac16dce9e28863cb7ee99d86362e30633aa77ab7a49d","family_id":"qf-mod40-binomial-expansion","seed":202611095038,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":12}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=12, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(12, [[0, 1, 12, 0], [1, 12, 11, 1], [2, 66, 10, 2], [3, 220, 9, 3], [4, 495, 8, 4], [5, 792, 7, 5], [6, 924, 6, 6], [7, 792, 5, 7], [8, 495, 4, 8], [9, 220, 3, 9], [10, 66, 2, 10], [11, 12, 1, 11], [12, 1, 0, 12]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (12, [[0, 1, 12, 0], [1, 12, 11, 1], [2, 66, 10, 2], [3, 220, 9, 3], [4, 495, 8, 4], [5, 792, 7, 5], [6, 924, 6, 6], [7, 792, 5, 7], [8, 495, 4, 8], [9, 220, 3, 9], [10, 66, 2, 10], [11, 12, 1, 11], [12, 1, 0, 12]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0039","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eec207fc0059dbdef010feebaa91faf18e4d967869b995e71fe9dd778dc1e5d9","family_id":"qf-mod40-binomial-expansion","seed":202611095039,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":13}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=13, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(13, [[0, 1, 13, 0], [1, 13, 12, 1], [2, 78, 11, 2], [3, 286, 10, 3], [4, 715, 9, 4], [5, 1287, 8, 5], [6, 1716, 7, 6], [7, 1716, 6, 7], [8, 1287, 5, 8], [9, 715, 4, 9], [10, 286, 3, 10], [11, 78, 2, 11], [12, 13, 1, 12], [13, 1, 0, 13]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (13, [[0, 1, 13, 0], [1, 13, 12, 1], [2, 78, 11, 2], [3, 286, 10, 3], [4, 715, 9, 4], [5, 1287, 8, 5], [6, 1716, 7, 6], [7, 1716, 6, 7], [8, 1287, 5, 8], [9, 715, 4, 9], [10, 286, 3, 10], [11, 78, 2, 11], [12, 13, 1, 12], [13, 1, 0, 13]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0040","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb267b9df094c9f841f151f97bdc090351cdc90a04eb650e045ed9a392c96c47","family_id":"qf-mod40-binomial-expansion","seed":202611095040,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":14}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=14, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(14, [[0, 1, 14, 0], [1, 14, 13, 1], [2, 91, 12, 2], [3, 364, 11, 3], [4, 1001, 10, 4], [5, 2002, 9, 5], [6, 3003, 8, 6], [7, 3432, 7, 7], [8, 3003, 6, 8], [9, 2002, 5, 9], [10, 1001, 4, 10], [11, 364, 3, 11], [12, 91, 2, 12], [13, 14, 1, 13], [14, 1, 0, 14]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (14, [[0, 1, 14, 0], [1, 14, 13, 1], [2, 91, 12, 2], [3, 364, 11, 3], [4, 1001, 10, 4], [5, 2002, 9, 5], [6, 3003, 8, 6], [7, 3432, 7, 7], [8, 3003, 6, 8], [9, 2002, 5, 9], [10, 1001, 4, 10], [11, 364, 3, 11], [12, 91, 2, 12], [13, 14, 1, 13], [14, 1, 0, 14]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0041","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bfa563ec1b2fafc76ab9cd5cbb24aa237986f1652cd69d30a97189b0a43b069","family_id":"qf-mod40-binomial-expansion","seed":202611095041,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":15}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=15, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(15, [[0, 1, 15, 0], [1, 15, 14, 1], [2, 105, 13, 2], [3, 455, 12, 3], [4, 1365, 11, 4], [5, 3003, 10, 5], [6, 5005, 9, 6], [7, 6435, 8, 7], [8, 6435, 7, 8], [9, 5005, 6, 9], [10, 3003, 5, 10], [11, 1365, 4, 11], [12, 455, 3, 12], [13, 105, 2, 13], [14, 15, 1, 14], [15, 1, 0, 15]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (15, [[0, 1, 15, 0], [1, 15, 14, 1], [2, 105, 13, 2], [3, 455, 12, 3], [4, 1365, 11, 4], [5, 3003, 10, 5], [6, 5005, 9, 6], [7, 6435, 8, 7], [8, 6435, 7, 8], [9, 5005, 6, 9], [10, 3003, 5, 10], [11, 1365, 4, 11], [12, 455, 3, 12], [13, 105, 2, 13], [14, 15, 1, 14], [15, 1, 0, 15]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0042","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02685b9f15f88fb114eb1a4aaceb471d861d68c564a776f9d2e9ea18895baf64","family_id":"qf-mod40-binomial-expansion","seed":202611095042,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":16}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=16, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(16, [[0, 1, 16, 0], [1, 16, 15, 1], [2, 120, 14, 2], [3, 560, 13, 3], [4, 1820, 12, 4], [5, 4368, 11, 5], [6, 8008, 10, 6], [7, 11440, 9, 7], [8, 12870, 8, 8], [9, 11440, 7, 9], [10, 8008, 6, 10], [11, 4368, 5, 11], [12, 1820, 4, 12], [13, 560, 3, 13], [14, 120, 2, 14], [15, 16, 1, 15], [16, 1, 0, 16]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (16, [[0, 1, 16, 0], [1, 16, 15, 1], [2, 120, 14, 2], [3, 560, 13, 3], [4, 1820, 12, 4], [5, 4368, 11, 5], [6, 8008, 10, 6], [7, 11440, 9, 7], [8, 12870, 8, 8], [9, 11440, 7, 9], [10, 8008, 6, 10], [11, 4368, 5, 11], [12, 1820, 4, 12], [13, 560, 3, 13], [14, 120, 2, 14], [15, 16, 1, 15], [16, 1, 0, 16]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0043","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7954fe81e2fb82b5bbd32a472856a7ef09b868c18f0a89003cd726f188fe99a","family_id":"qf-mod40-binomial-expansion","seed":202611095043,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":17}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=17, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(17, [[0, 1, 17, 0], [1, 17, 16, 1], [2, 136, 15, 2], [3, 680, 14, 3], [4, 2380, 13, 4], [5, 6188, 12, 5], [6, 12376, 11, 6], [7, 19448, 10, 7], [8, 24310, 9, 8], [9, 24310, 8, 9], [10, 19448, 7, 10], [11, 12376, 6, 11], [12, 6188, 5, 12], [13, 2380, 4, 13], [14, 680, 3, 14], [15, 136, 2, 15], [16, 17, 1, 16], [17, 1, 0, 17]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (17, [[0, 1, 17, 0], [1, 17, 16, 1], [2, 136, 15, 2], [3, 680, 14, 3], [4, 2380, 13, 4], [5, 6188, 12, 5], [6, 12376, 11, 6], [7, 19448, 10, 7], [8, 24310, 9, 8], [9, 24310, 8, 9], [10, 19448, 7, 10], [11, 12376, 6, 11], [12, 6188, 5, 12], [13, 2380, 4, 13], [14, 680, 3, 14], [15, 136, 2, 15], [16, 17, 1, 16], [17, 1, 0, 17]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0044","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"835dcca8d7245fb58c69a7e9efec5405869da760172c93091899b2460c35e5f9","family_id":"qf-mod40-binomial-expansion","seed":202611095044,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":18}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=18, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(18, [[0, 1, 18, 0], [1, 18, 17, 1], [2, 153, 16, 2], [3, 816, 15, 3], [4, 3060, 14, 4], [5, 8568, 13, 5], [6, 18564, 12, 6], [7, 31824, 11, 7], [8, 43758, 10, 8], [9, 48620, 9, 9], [10, 43758, 8, 10], [11, 31824, 7, 11], [12, 18564, 6, 12], [13, 8568, 5, 13], [14, 3060, 4, 14], [15, 816, 3, 15], [16, 153, 2, 16], [17, 18, 1, 17], [18, 1, 0, 18]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (18, [[0, 1, 18, 0], [1, 18, 17, 1], [2, 153, 16, 2], [3, 816, 15, 3], [4, 3060, 14, 4], [5, 8568, 13, 5], [6, 18564, 12, 6], [7, 31824, 11, 7], [8, 43758, 10, 8], [9, 48620, 9, 9], [10, 43758, 8, 10], [11, 31824, 7, 11], [12, 18564, 6, 12], [13, 8568, 5, 13], [14, 3060, 4, 14], [15, 816, 3, 15], [16, 153, 2, 16], [17, 18, 1, 17], [18, 1, 0, 18]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0045","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d34617516997dad1199c17c143843c3297c8e9aabead1deca0bf91def62d392","family_id":"qf-mod40-binomial-expansion","seed":202611095045,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":19}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=19, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(19, [[0, 1, 19, 0], [1, 19, 18, 1], [2, 171, 17, 2], [3, 969, 16, 3], [4, 3876, 15, 4], [5, 11628, 14, 5], [6, 27132, 13, 6], [7, 50388, 12, 7], [8, 75582, 11, 8], [9, 92378, 10, 9], [10, 92378, 9, 10], [11, 75582, 8, 11], [12, 50388, 7, 12], [13, 27132, 6, 13], [14, 11628, 5, 14], [15, 3876, 4, 15], [16, 969, 3, 16], [17, 171, 2, 17], [18, 19, 1, 18], [19, 1, 0, 19]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (19, [[0, 1, 19, 0], [1, 19, 18, 1], [2, 171, 17, 2], [3, 969, 16, 3], [4, 3876, 15, 4], [5, 11628, 14, 5], [6, 27132, 13, 6], [7, 50388, 12, 7], [8, 75582, 11, 8], [9, 92378, 10, 9], [10, 92378, 9, 10], [11, 75582, 8, 11], [12, 50388, 7, 12], [13, 27132, 6, 13], [14, 11628, 5, 14], [15, 3876, 4, 15], [16, 969, 3, 16], [17, 171, 2, 17], [18, 19, 1, 18], [19, 1, 0, 19]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0046","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0aedc127c9cafb8e44db0fe7320d1c84a2dad136d25e45209eefe470e64a538","family_id":"qf-mod40-binomial-expansion","seed":202611095046,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":20}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=20, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(20, [[0, 1, 20, 0], [1, 20, 19, 1], [2, 190, 18, 2], [3, 1140, 17, 3], [4, 4845, 16, 4], [5, 15504, 15, 5], [6, 38760, 14, 6], [7, 77520, 13, 7], [8, 125970, 12, 8], [9, 167960, 11, 9], [10, 184756, 10, 10], [11, 167960, 9, 11], [12, 125970, 8, 12], [13, 77520, 7, 13], [14, 38760, 6, 14], [15, 15504, 5, 15], [16, 4845, 4, 16], [17, 1140, 3, 17], [18, 190, 2, 18], [19, 20, 1, 19], [20, 1, 0, 20]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (20, [[0, 1, 20, 0], [1, 20, 19, 1], [2, 190, 18, 2], [3, 1140, 17, 3], [4, 4845, 16, 4], [5, 15504, 15, 5], [6, 38760, 14, 6], [7, 77520, 13, 7], [8, 125970, 12, 8], [9, 167960, 11, 9], [10, 184756, 10, 10], [11, 167960, 9, 11], [12, 125970, 8, 12], [13, 77520, 7, 13], [14, 38760, 6, 14], [15, 15504, 5, 15], [16, 4845, 4, 16], [17, 1140, 3, 17], [18, 190, 2, 18], [19, 20, 1, 19], [20, 1, 0, 20]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0047","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38fcaa28ce86999a50dea4366babdfeee8010274dc252d7b8c10a72b2cb04b78","family_id":"qf-mod40-binomial-expansion","seed":202611095047,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":21}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=21, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(21, [[0, 1, 21, 0], [1, 21, 20, 1], [2, 210, 19, 2], [3, 1330, 18, 3], [4, 5985, 17, 4], [5, 20349, 16, 5], [6, 54264, 15, 6], [7, 116280, 14, 7], [8, 203490, 13, 8], [9, 293930, 12, 9], [10, 352716, 11, 10], [11, 352716, 10, 11], [12, 293930, 9, 12], [13, 203490, 8, 13], [14, 116280, 7, 14], [15, 54264, 6, 15], [16, 20349, 5, 16], [17, 5985, 4, 17], [18, 1330, 3, 18], [19, 210, 2, 19], [20, 21, 1, 20], [21, 1, 0, 21]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (21, [[0, 1, 21, 0], [1, 21, 20, 1], [2, 210, 19, 2], [3, 1330, 18, 3], [4, 5985, 17, 4], [5, 20349, 16, 5], [6, 54264, 15, 6], [7, 116280, 14, 7], [8, 203490, 13, 8], [9, 293930, 12, 9], [10, 352716, 11, 10], [11, 352716, 10, 11], [12, 293930, 9, 12], [13, 203490, 8, 13], [14, 116280, 7, 14], [15, 54264, 6, 15], [16, 20349, 5, 16], [17, 5985, 4, 17], [18, 1330, 3, 18], [19, 210, 2, 19], [20, 21, 1, 20], [21, 1, 0, 21]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0048","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"322148fcbbd558ebcdb0e8488271b3b1160106daff85a692ccf064be64afe5d3","family_id":"qf-mod40-binomial-expansion","seed":202611095048,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":22}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=22, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(22, [[0, 1, 22, 0], [1, 22, 21, 1], [2, 231, 20, 2], [3, 1540, 19, 3], [4, 7315, 18, 4], [5, 26334, 17, 5], [6, 74613, 16, 6], [7, 170544, 15, 7], [8, 319770, 14, 8], [9, 497420, 13, 9], [10, 646646, 12, 10], [11, 705432, 11, 11], [12, 646646, 10, 12], [13, 497420, 9, 13], [14, 319770, 8, 14], [15, 170544, 7, 15], [16, 74613, 6, 16], [17, 26334, 5, 17], [18, 7315, 4, 18], [19, 1540, 3, 19], [20, 231, 2, 20], [21, 22, 1, 21], [22, 1, 0, 22]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (22, [[0, 1, 22, 0], [1, 22, 21, 1], [2, 231, 20, 2], [3, 1540, 19, 3], [4, 7315, 18, 4], [5, 26334, 17, 5], [6, 74613, 16, 6], [7, 170544, 15, 7], [8, 319770, 14, 8], [9, 497420, 13, 9], [10, 646646, 12, 10], [11, 705432, 11, 11], [12, 646646, 10, 12], [13, 497420, 9, 13], [14, 319770, 8, 14], [15, 170544, 7, 15], [16, 74613, 6, 16], [17, 26334, 5, 17], [18, 7315, 4, 18], [19, 1540, 3, 19], [20, 231, 2, 20], [21, 22, 1, 21], [22, 1, 0, 22]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0049","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"337cdef8e30d1b61e5c18099d332cbbd2596bbc0ceed96c18ad6f7ee28afe5a1","family_id":"qf-mod40-binomial-expansion","seed":202611095049,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":23}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=23, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(23, [[0, 1, 23, 0], [1, 23, 22, 1], [2, 253, 21, 2], [3, 1771, 20, 3], [4, 8855, 19, 4], [5, 33649, 18, 5], [6, 100947, 17, 6], [7, 245157, 16, 7], [8, 490314, 15, 8], [9, 817190, 14, 9], [10, 1144066, 13, 10], [11, 1352078, 12, 11], [12, 1352078, 11, 12], [13, 1144066, 10, 13], [14, 817190, 9, 14], [15, 490314, 8, 15], [16, 245157, 7, 16], [17, 100947, 6, 17], [18, 33649, 5, 18], [19, 8855, 4, 19], [20, 1771, 3, 20], [21, 253, 2, 21], [22, 23, 1, 22], [23, 1, 0, 23]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (23, [[0, 1, 23, 0], [1, 23, 22, 1], [2, 253, 21, 2], [3, 1771, 20, 3], [4, 8855, 19, 4], [5, 33649, 18, 5], [6, 100947, 17, 6], [7, 245157, 16, 7], [8, 490314, 15, 8], [9, 817190, 14, 9], [10, 1144066, 13, 10], [11, 1352078, 12, 11], [12, 1352078, 11, 12], [13, 1144066, 10, 13], [14, 817190, 9, 14], [15, 490314, 8, 15], [16, 245157, 7, 16], [17, 100947, 6, 17], [18, 33649, 5, 18], [19, 8855, 4, 19], [20, 1771, 3, 20], [21, 253, 2, 21], [22, 23, 1, 22], [23, 1, 0, 23]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0050","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24e1908ee45ca42532415665075d5aa9f8121ea581f44ae6a36072f7f8a38280","family_id":"qf-mod40-binomial-expansion","seed":202611095050,"semantic_template":"qf-mod40-binomial-expansion/{\"n\":24}"},"fields":{"title":"The Finite Expansion Includes Every k from Zero through n","stem":"Use natural exponent n=24, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).","answer":{"kind":"expression","value":"(24, [[0, 1, 24, 0], [1, 24, 23, 1], [2, 276, 22, 2], [3, 2024, 21, 3], [4, 10626, 20, 4], [5, 42504, 19, 5], [6, 134596, 18, 6], [7, 346104, 17, 7], [8, 735471, 16, 8], [9, 1307504, 15, 9], [10, 1961256, 14, 10], [11, 2496144, 13, 11], [12, 2704156, 12, 12], [13, 2496144, 11, 13], [14, 1961256, 10, 14], [15, 1307504, 9, 15], [16, 735471, 8, 16], [17, 346104, 7, 17], [18, 134596, 6, 18], [19, 42504, 5, 19], [20, 10626, 4, 20], [21, 2024, 3, 21], [22, 276, 2, 22], [23, 24, 1, 23], [24, 1, 0, 24]], complete_k_zero_through_n_coefficients)","explanation":"Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (24, [[0, 1, 24, 0], [1, 24, 23, 1], [2, 276, 22, 2], [3, 2024, 21, 3], [4, 10626, 20, 4], [5, 42504, 19, 5], [6, 134596, 18, 6], [7, 346104, 17, 7], [8, 735471, 16, 8], [9, 1307504, 15, 9], [10, 1961256, 14, 10], [11, 2496144, 13, 11], [12, 2704156, 12, 12], [13, 2496144, 11, 13], [14, 1961256, 10, 14], [15, 1307504, 9, 15], [16, 735471, 8, 16], [17, 346104, 7, 17], [18, 134596, 6, 18], [19, 42504, 5, 19], [20, 10626, 4, 20], [21, 2024, 3, 21], [22, 276, 2, 22], [23, 24, 1, 23], [24, 1, 0, 24]], complete_k_zero_through_n_coefficients)."},"skills":["The Finite Expansion Includes Every k from Zero through n"],"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-20261014-advance-20261109-r05-binomial-theorem-0051","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99265716d3d414031f651aae979af53122abc6f6ea96074e51c41672e33127e5","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095051,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":0,\"x\":-3,\"y\":2}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=0, exact commuting integers x=-3, y=2. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(0, -3, 2, 1, 1, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (0, -3, 2, 1, 1, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0052","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"687e2dd4911f44345ddd72631eb10b85a010da7215777531a4c3cca53d699dcf","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095052,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":1,\"x\":-2,\"y\":1}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=1, exact commuting integers x=-2, y=1. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(1, -2, 1, -1, -1, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (1, -2, 1, -1, -1, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0053","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30a22018fcb38373d605a20e0eaa2d8c7437720339ccb5c3c51110284c338d8f","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095053,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":2,\"x\":-1,\"y\":0}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=2, exact commuting integers x=-1, y=0. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(2, -1, 0, 1, 1, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (2, -1, 0, 1, 1, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0054","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34daf4735dd9f11accad0762962ffcb1729028e6d4198df3a0205b7a42a81144","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095054,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":3,\"x\":0,\"y\":-1}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=3, exact commuting integers x=0, y=-1. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(3, 0, -1, -1, -1, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (3, 0, -1, -1, -1, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0055","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84f3b71d41cbc0762b3c9939fe4719cb684d0859e32b40b7b14c6162ddaacc2b","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095055,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":4,\"x\":1,\"y\":-2}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=4, exact commuting integers x=1, y=-2. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(4, 1, -2, 1, 1, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (4, 1, -2, 1, 1, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0056","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cb0a0a23948d7554ae65952fa69ad883d1d2a601487f97048ea848be36d9822","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095056,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":5,\"x\":2,\"y\":2}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=5, exact commuting integers x=2, y=2. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(5, 2, 2, 1024, 1024, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (5, 2, 2, 1024, 1024, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0057","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1affa3cdfc0b770d93a6c2d4d35d3ff583a73f067247ffc7f061b1841c1bd78c","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095057,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":6,\"x\":3,\"y\":1}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=6, exact commuting integers x=3, y=1. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(6, 3, 1, 4096, 4096, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (6, 3, 1, 4096, 4096, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0058","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1d6ba36610032034746748fdc8cb00b7f4ba99e5d014cb3d3782cfac6d913b9","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095058,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":7,\"x\":-3,\"y\":0}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=7, exact commuting integers x=-3, y=0. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(7, -3, 0, -2187, -2187, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (7, -3, 0, -2187, -2187, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0059","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b64e16b5d2d9c07d534b43ee23698ac4c54b2c8e092dc18793fc246567e1ee5","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095059,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":8,\"x\":-2,\"y\":-1}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=8, exact commuting integers x=-2, y=-1. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(8, -2, -1, 6561, 6561, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (8, -2, -1, 6561, 6561, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0060","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43a78e4e1a5006b3f2b19f290fff7b75747e65c1f57cb3910e216ceb0436ebfc","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095060,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":9,\"x\":-1,\"y\":-2}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=9, exact commuting integers x=-1, y=-2. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(9, -1, -2, -19683, -19683, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (9, -1, -2, -19683, -19683, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0061","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6766ab6818646eaa98c8d0dd1e4ecfed49267b24326bc932a81a6320030ce10f","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095061,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":10,\"x\":0,\"y\":2}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=10, exact commuting integers x=0, y=2. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(10, 0, 2, 1024, 1024, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (10, 0, 2, 1024, 1024, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0062","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af4154aa97908dfe046b24b618a06af6ad3092932c86e5aec770625b284db047","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095062,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":11,\"x\":1,\"y\":1}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=11, exact commuting integers x=1, y=1. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(11, 1, 1, 2048, 2048, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (11, 1, 1, 2048, 2048, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0063","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a307ac1992e111786aa4bbf7ed7e0911708371a9493255da1c9335d571a62e5","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095063,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":12,\"x\":2,\"y\":0}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=12, exact commuting integers x=2, y=0. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(12, 2, 0, 4096, 4096, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (12, 2, 0, 4096, 4096, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0064","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68709557c411421e722a54bd30726f0d1ffeef1171c77dc814885ac4db5baf81","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095064,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":13,\"x\":3,\"y\":-1}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=13, exact commuting integers x=3, y=-1. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(13, 3, -1, 8192, 8192, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (13, 3, -1, 8192, 8192, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0065","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ba88213bc1ea014678a55b12aaf4565366397e54e0a35a95c04672ccaf45d28","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095065,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":14,\"x\":-3,\"y\":-2}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=14, exact commuting integers x=-3, y=-2. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(14, -3, -2, 6103515625, 6103515625, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (14, -3, -2, 6103515625, 6103515625, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0066","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bf27582977bcc4a12ddbfe9d88e761188c39892d606e39a0d4bc0eb2d1a0180","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095066,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":15,\"x\":-2,\"y\":2}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=15, exact commuting integers x=-2, y=2. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(15, -2, 2, 0, 0, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (15, -2, 2, 0, 0, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0067","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e76bf4d73aae3ea395370c55d75b21b65501c480c1c93ab6b738ded5fb960e3","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095067,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":16,\"x\":-1,\"y\":1}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=16, exact commuting integers x=-1, y=1. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(16, -1, 1, 0, 0, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (16, -1, 1, 0, 0, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0068","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4d04cc67d8f4224f6aa02a5bca445c22897f247698da9a9655cf4e30da054c0","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095068,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":17,\"x\":0,\"y\":0}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=17, exact commuting integers x=0, y=0. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(17, 0, 0, 0, 0, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (17, 0, 0, 0, 0, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0069","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"042146ca1652daecd014719165896b50b7834f852d12837404f791ac3400d267","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095069,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":18,\"x\":1,\"y\":-1}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=18, exact commuting integers x=1, y=-1. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(18, 1, -1, 0, 0, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (18, 1, -1, 0, 0, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0070","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8000045f8dc6dc3cfd8a26cac391556fdf3c2a68a023a0209baa1a323de48872","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095070,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":19,\"x\":2,\"y\":-2}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=19, exact commuting integers x=2, y=-2. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(19, 2, -2, 0, 0, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (19, 2, -2, 0, 0, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0071","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e445a34c7f452364200013b74ea3a5e97ae54c33fb9f9bcd33becf032585c11e","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095071,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":20,\"x\":3,\"y\":2}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=20, exact commuting integers x=3, y=2. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(20, 3, 2, 95367431640625, 95367431640625, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (20, 3, 2, 95367431640625, 95367431640625, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0072","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e72fc10a85582e34722a507e146f0c3a3ffd52da34143214e313e32084615f00","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095072,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":21,\"x\":-3,\"y\":1}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=21, exact commuting integers x=-3, y=1. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(21, -3, 1, -2097152, -2097152, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (21, -3, 1, -2097152, -2097152, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0073","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a43d8fd7a7301f1cc5c2b7c50d7baeef29e7b4e7d9d7e2018619fb9bb88e286","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095073,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":22,\"x\":-2,\"y\":0}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=22, exact commuting integers x=-2, y=0. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(22, -2, 0, 4194304, 4194304, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (22, -2, 0, 4194304, 4194304, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0074","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ea820d0936be7dbdde412a2cba320a9b5bd0e7ea5d13183d53a25938dfeefa6","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095074,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":23,\"x\":-1,\"y\":-1}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=23, exact commuting integers x=-1, y=-1. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(23, -1, -1, -8388608, -8388608, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (23, -1, -1, -8388608, -8388608, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0075","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b36867bb0f1accc84bfd1f2fa5a5d3ac8e009714bb572d00b323afbbab1c7ee7","family_id":"qf-mod40-binomial-evaluation-identity","seed":202611095075,"semantic_template":"qf-mod40-binomial-evaluation-identity/{\"n\":24,\"x\":0,\"y\":-2}"},"fields":{"title":"Compare the Whole Binomial Sum with an Exact Integer Power","stem":"Use natural exponent n=24, exact commuting integers x=0, y=-2. Give (n, x, y, power, expansion_sum, equal).","answer":{"kind":"expression","value":"(24, 0, -2, 16777216, 16777216, Yes)","explanation":"Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (24, 0, -2, 16777216, 16777216, Yes)."},"skills":["Compare the Whole Binomial Sum with an Exact Integer Power"],"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-20261014-advance-20261109-r05-binomial-theorem-0076","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58467e0ccadcc3715fffc43b40591e4f7e863763236f301bd2cdee614c287130","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095076,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-1}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-1. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-1, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-1, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0077","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb82620df0e199ee26edc36bfdc3ae44589deb2e7d7b0c4e421f398cf9df0e8d","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095077,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-2}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-2. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-2, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-2, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0078","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f78c4a981b9f2ef51ca0820be4182c349eb808acf147e9d69deaeb059dfae976","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095078,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-3}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-3. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-3, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-3, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0079","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"add1488a26d514fe4f174fd16a02823bf663745a1639e0783d55c1c0da0100e7","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095079,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-4}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-4. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-4, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-4, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0080","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"457f96d53e1a228c374fb6e565a7fa7dff4de3be8532706a79746c54faba5234","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095080,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-5}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-5. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-5, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-5, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0081","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ba4c9068af23a455e795166e3209d222540f15c93d6ada4e3403e17b77d68a3","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095081,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-6}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-6. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-6, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-6, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0082","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0e9f3dc625b3cd47a8d19a1239b86f05441ab2088a5468dd1ba738cd97aa6e8","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095082,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-7}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-7. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-7, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-7, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0083","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"555c6ee6b2c5f774c438d790221953eb789b95d0ecf5fe4ca9aa1d9d96687db7","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095083,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-8}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-8. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-8, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-8, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0084","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fbc51601b2cf362c9d362963257602507bb40bbb67ce951d978b8764e368102","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095084,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-9}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-9. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-9, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-9, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0085","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4fa706fea23b3d67e00aba7206d9837d515362f84a3c1381be82f5bfd74cd62","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095085,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-10}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-10. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-10, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-10, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0086","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70bc6617d66b25cb29f75ffb83e5af75cdec836882e1c5d41f8706ebdce8aa19","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095086,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-11}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-11. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-11, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-11, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0087","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f191819349c570eb7cabf6576c1c25a5d983d956778f94deb98ad3bda899ac3","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095087,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-12}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-12. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-12, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-12, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0088","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a0b29d9e988409d686c64204d13e7ea50591c3ffa86d4a9ed83a5e475c9128f","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095088,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-13}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-13. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-13, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-13, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0089","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7d2f53ad45d026374932f449aebf6619bf95c5093daa3f3305adc7d1721fcaa","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095089,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-14}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-14. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-14, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-14, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0090","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1a0c756d84cc9f5955b25ce6cad34d3b7c66e829b50f8d2630423d4b9f005ca","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095090,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-15}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-15. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-15, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-15, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0091","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb9c5a9556c893c66c0541e290df8f3581ec72638e4c69a2e03707648e7992b7","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095091,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-16}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-16. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-16, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-16, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0092","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3641e862bb44ab3da472c50743b54413ee3c0d293eb9c58d8e9c2fe575bc7b46","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095092,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-17}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-17. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-17, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-17, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0093","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03396ac9a8c5ebca524b1b80294e84a353c1c95a55d3280f2420bf2909c6a9e1","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095093,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-18}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-18. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-18, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-18, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0094","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01a547b8e80d00c5eb3d1968d8adbb1faee28bb07be7d4b68c54263e7f6c6d6a","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095094,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-19}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-19. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-19, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-19, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0095","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dee03ed03aa4162580cd6f6fb7399959b30f986d24c42968f8af7ff8ee0aea9f","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095095,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-20}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-20. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-20, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-20, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0096","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c01f03607beb18a0d4ca5ae1b2687b4d608eefeb47c731126767cb6d9f71e156","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095096,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-21}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-21. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-21, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-21, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0097","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17a4cf16981f03d18334dbed079317fe0f1484996ac9b419703e924938c66a62","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095097,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-22}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-22. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-22, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-22, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0098","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13287ac53cd3772fb9bd73f01825e498498a9e14fe825e910d70116f8908bb38","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095098,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-23}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-23. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-23, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-23, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0099","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acbd928d78ac1ec576350ed1e927ac9852b3f283a430d2e1772f2370365bc7b8","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095099,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-24}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-24. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-24, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-24, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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-20261014-advance-20261109-r05-binomial-theorem-0100","batch_id":"batch-20261014-advance-20261109-r05-binomial-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38e173d8e63e8472cc7879d91a16e09e958596f0297a5a63678f936083730636","family_id":"qf-mod40-binomial-exponent-boundary","seed":202611095100,"semantic_template":"qf-mod40-binomial-exponent-boundary/{\"exponent\":-25}"},"fields":{"title":"Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem","stem":"Use boundary exponent=-25. Give (exponent, finite_polynomial_claim, boundary).","answer":{"kind":"expression","value":"(-25, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)","explanation":"The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-25, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)."},"skills":["Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem"],"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."}}