{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261109-r05-binomial-theorem-0006", "pilot_number": "P1000-0406", "number": 406, "family_id": "qf-mod40-binomial-coefficient", "title": { "zh-Hant": "第 406 題：二項式定理", "en": "Question 406: Exact Binomial Coefficients Keep k and n Minus k Distinct" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261109-r05-binomial-theorem-0006", "batch_id": "batch-20261014-advance-20261109-r05-binomial-theorem-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-binomial-theorem" ], "title_zh": "二項式定理", "stem_zh": "commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":5,\"k\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(5, 5, 1, 1, 是)", "explanation_zh": "Wave6第三層核對=(5, 5, 1, 1, 是)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "二項式定理" ] }, "variant": { "family_id": "qf-mod40-binomial-coefficient", "seed": 202611095006, "parameters": { "n": 5, "k": 5, "domain": "Exact_commutative_binomial_theorem", "operation": "binomial-coefficient", "semantic_template": "qf-mod40-binomial-coefficient/{\"k\":5,\"n\":5}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "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." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "commutative coefficient setting與n in natural including zero。操作=binomial-coefficient；資料={\"n\":5,\"k\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "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)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "n=5≥0; multiplication commutes", "after": "group the finite product by the number of y choices", "relation": "implies", "rule_id": "commutative_product", "conditions": [], "explanation": { "zh-Hant": "每個因子選x或y，依選y的次數分組，係數先作整數計數。", "en": "Choose x or y in each factor and group by the number of y choices; count coefficients as integers first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "choose 5 positions out of 5", "after": "C(5,5)=120/(120·1)=1", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "5", "5" ], "output": "1", "label": "C(5,5)" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "choose 0 positions out of 5", "after": "C(5,0)=120/(1·120)=1", "relation": "implies", "rule_id": "subset_count", "conditions": [], "explanation": { "zh-Hant": "選取 k 個位置的每個子集只計一次；除掉兩組內部排列後得到整數係數。", "en": "Count each k-position subset once, dividing out reorderings inside each of the two groups." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": { "operation": "binomial_count", "inputs": [ "5", "0" ], "output": "1", "label": "C(5,0)" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "choose subset of size 5", "after": "its complement has size 0", "relation": "implies", "rule_id": "subset_complement_bijection", "conditions": [], "explanation": { "zh-Hant": "取補集是互逆映射，因此這兩個選法數目相同。", "en": "Taking complements is an invertible map, so the two counts agree." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "C(5,5)=1; C(5,0)=1", "after": "symmetric coefficients equal: 是", "relation": "implies", "rule_id": "coefficient_symmetry", "conditions": [], "explanation": { "zh-Hant": "數值核對符合補集雙射；n=0仍只有空子集一種。", "en": "The values agree with the complement bijection; n=0 still has exactly the empty subset." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": null } ], "math_model": { "result": [ 5, 5, 1, 1, true ] }, "conclusion": { "canonical_value": "(5, 5, 1, 1, 是)", "explanation": { "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。", "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings." } }, "definition_note": { "zh-Hant": "0≤k≤n 的整數計數；C(n,k) 為選 k 個位置的子集數，補集雙射給出對稱性。", "en": "Integer counting with 0≤k≤n. C(n,k) counts k-position subsets; the complement bijection gives symmetry." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261109-r05-binomial-theorem-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261109-r05-binomial-theorem-foundations/questions.jsonl", "batch_sha256": "f83302c1353681ecb1209ee1ccbef0d266f15fb9415191ed9289eb329e1175cc", "original_line": 6, "original_line_sha256": "1e8256bad9ce171d8049124c69e232b60373a688e762263d8e07867264c3e8a7", "question_compact_sha256": "c24e12a4dcbec40c245eadeb24b486c0f40209bb2ee42fde8ac65490f358a13e", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261109-r05-binomial-theorem-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261109-r05-binomial-theorem-foundations.en.jsonl", "english_row_compact_sha256": "5833d7fd4b2d6d5cde43db07a2690e7c545db49ce08e0770b5bf658d218920ad" }, "mko_bindings": { "mko-binomial-theorem": { "sha256": "c8fefe3783fc4f7633b4b22b8cf149d96a683900f181e3b42c690409ce90e5b7", "english_sha256": "1086dfe7c31f4e6bf8934681a0a080b50c14cb828f1a8b17e59143ec2cf06c83", "original_path": "public/data/mko/binomial-theorem.json", "english_path": "public/data/i18n/editions/mko-binomial-theorem.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "afa7c0e9f7084cc5929124dcffe9929d5db7def32e40047056d19ea25535873e", "original_candidate_sha256": "5cad36a637f2f5dde9c1f83596f957475484508cf0194d4870834d4f4f584de3", "original_candidate_line_sha256": "5cad36a637f2f5dde9c1f83596f957475484508cf0194d4870834d4f4f584de3" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。", "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI review or general formal proof. Original questions, answers, MKOs, formulas, ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-003-01", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772", "question_first_witness_sha256": "77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "780f9a9eb0e5948ad119c257245c15acaed75c425ad2acd84d1a045e08bff8db", "arithmetic_steps": 2 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }