{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20260930-r06-permutation-combination-quotient-0007", "pilot_number": "R0067-34157", "number": 34157, "family_id": "qf-permutation-combination-quotient", "title": { "zh-Hant": "第 34157 題：排列與組合的商", "en": "Question 34157: Quotient of Permutation and Combination Counts" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20260930-r06-permutation-combination-quotient-0007", "batch_id": "batch-20260930-r06-permutation-combination-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-permutation-combination" ], "title_zh": "排列與組合的商", "stem_zh": "從 4 個彼此相異物件無放回且不重複選取 2 個，空選取計一種。P(n,k) 表示有序數、C(n,k) 表示無序數；求 P(n,k)/C(n,k) 的精確整數值。", "answer": { "kind": "short_text", "value": "2", "explanation_zh": "每個 k 元子集合有 k! 種排序，因此此商為 2；空集合也恰有一個排序。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "每個子集合的排序", "空選取" ] }, "variant": { "family_id": "qf-permutation-combination-quotient", "seed": 202609306307, "parameters": { "operation": "quotient", "n": 4, "k": 2, "semantic_template": "qf-permutation-combination-quotient/{\"k\":2,\"n\":4,\"operation\":\"quotient\"}" } }, "verification": { "status": "candidate", "checks": [ "json_schema", "producer_local_answer_invariants", "explanation_witness_accuracy" ], "warning_zh": "本題保持 candidate；有限精確計算與產生者自檢不構成人工審查、形式證明或主架構者驗收。" }, "provenance": { "generator_role": "question_producer", "generator_model": "Codex GPT-6 question producer", "generated_at": "2026-09-30", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20260930-r06-permutation-combination-quotient-0007", "batch_id": "batch-20260930-r06-permutation-combination-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3de3b845d21257b1154e65d6906b03580db6c65333a8693f4a6736f69ee8aee5", "family_id": "qf-permutation-combination-quotient", "seed": 202609306307, "semantic_template": "qf-permutation-combination-quotient/{\"k\":2,\"n\":4,\"operation\":\"quotient\"}" }, "fields": { "title": "Quotient of Permutation and Combination Counts", "stem": "For selection of k=2 distinct objects without replacement from n=4, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.", "answer": { "kind": "short_text", "value": "2", "explanation": "Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering." }, "skills": [ "orderings of each subset", "empty selection" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "從 4 個彼此相異物件無放回且不重複選取 2 個，空選取計一種。P(n,k) 表示有序數、C(n,k) 表示無序數；求 P(n,k)/C(n,k) 的精確整數值。", "en": "For selection of k=2 distinct objects without replacement from n=4, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-permutation-combination-quotient, [k, n, operation]]", "after": "derived = [quotient, no-invented-domain]", "relation": "implies", "rule_id": "R67_SOURCE_specific_keys_operation_domain_and_numeric_bank", "conditions": [], "explanation": { "zh-Hant": "按本bank核keys與operation/domain：legacy為各表underscore，MOD50用local hyphen；BASE參數較寬不加未知欄限制。Number.isInteger與SAFE兩種來源界分開，原JSON數字表示仍完整保存。", "en": "Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [4, 2]", "after": "derived = [[4, 3], 1]", "relation": "implies", "rule_id": "Selection_DISTINCT_no_replacement_order_and_EMPTY_convention", "conditions": [], "explanation": { "zh-Hant": "n個不同物件無重複無放回，ordered selection第i位置有n−i+1個選擇。k=0空選取恰1種，乘法從1開始；本bank到n=200，不搬其他bank的32或64界。", "en": "Choose ordered positions from n DISTINCT objects WITHOUT replacement.Position i has n−i+1 choices.For k=0,the empty selection counts ONCE and the product starts at1.THIS bank permits n≤200,not another bank’s32/64 cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Permutation ORIGINAL available choices position1 := mul(1, 4)", "after": "Permutation ORIGINAL available choices position1 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Permutation ORIGINAL available choices position1，不以小數近似替代。", "en": "Recompute Permutation ORIGINAL available choices position1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "4" ], "output": "4", "label": "Permutation ORIGINAL available choices position1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Permutation ORIGINAL available choices position2 := mul(4, 3)", "after": "Permutation ORIGINAL available choices position2 = 12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Permutation ORIGINAL available choices position2，不以小數近似替代。", "en": "Recompute Permutation ORIGINAL available choices position2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "3" ], "output": "12", "label": "Permutation ORIGINAL available choices position2" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Quotient ORIGINAL factorial factor1 := mul(1, 1)", "after": "Quotient ORIGINAL factorial factor1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Quotient ORIGINAL factorial factor1，不以小數近似替代。", "en": "Recompute Quotient ORIGINAL factorial factor1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Quotient ORIGINAL factorial factor1" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Quotient ORIGINAL factorial factor2 := mul(1, 2)", "after": "Quotient ORIGINAL factorial factor2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Quotient ORIGINAL factorial factor2，不以小數近似替代。", "en": "Recompute Quotient ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Quotient ORIGINAL factorial factor2" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Quotient ORIGINAL permutations divided by subset orderings := div(12, 2)", "after": "Quotient ORIGINAL permutations divided by subset orderings = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Quotient ORIGINAL permutations divided by subset orderings，不以小數近似替代。", "en": "Recompute Quotient ORIGINAL permutations divided by subset orderings with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "12", "2" ], "output": "6", "label": "Quotient ORIGINAL permutations divided by subset orderings" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Quotient ORIGINAL permutations over combinations := div(12, 6)", "after": "Quotient ORIGINAL permutations over combinations = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Quotient ORIGINAL permutations over combinations，不以小數近似替代。", "en": "Recompute Quotient ORIGINAL permutations over combinations with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "12", "6" ], "output": "2", "label": "Quotient ORIGINAL permutations over combinations" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [4, 2, 12, 6]", "after": "derived = [2, 2]", "relation": "implies", "rule_id": "Selection_EVERY_k_subset_has_k_factorial_orderings", "conditions": [], "explanation": { "zh-Hant": "每個k元素subset恰有k!種次序，ordered selections依其底層subset分成等大fibers，故P=C*k!且C>0，P/C=k!。k=0仍1/1=1，不是估計比率或機率。", "en": "EVERY k-element subset has exactly k! orderings.Ordered selections partition into equal fibers over their underlying subsets,so P=C*k!,C>0,and P/C=k!.At k=0 this remains1/1=1,not an estimated ratio or probability." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null } ], "math_model": { "result": "2" }, "conclusion": { "canonical_value": "2", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20260930-r06-permutation-combination-foundations", "batch_path": "public/data/questions/batches/batch-20260930-r06-permutation-combination-foundations/questions.jsonl", "batch_sha256": "e40d1692407a43376f6370aae8b40042e482d2017d4ad08bd18bdf6694032044", "original_line": 82, "original_line_sha256": "87803683e326887a5023f253ec293ee2d19d56b4699b78745d4a22370f281959", "question_compact_sha256": "3de3b845d21257b1154e65d6906b03580db6c65333a8693f4a6736f69ee8aee5", "english_row_compact_sha256": "71b1ea833110140fd4bc55d21881cbb22dfaa2635ff6a3b5d3f81cbe63511bcd", "english_manifest_path": "public/data/i18n/editions/batch-20260930-r06-permutation-combination-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20260930-r06-permutation-combination-foundations.en.jsonl" }, "mko_bindings": { "mko-permutation-combination": { "sha256": "bd1ee61b3eaee90e08b3bb7a753128635ac36421438a78c41bba6dc768077c96", "english_sha256": "d86418363f341c4360d50617abf179e87c404fcc7d85b191d8da6e74d8bb6200", "original_path": "public/data/mko/permutation-combination.json", "english_path": "public/data/i18n/editions/mko-permutation-combination.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "6bb156cec7bdb6baf2497762702fc01f94fd1db55a5926bc575b12ff9237a4de", "original_candidate_sha256": "5ea24ed0d02357ed808bd425c88e3fd0999519efc31e2b6f9c820a0025ad9c3e", "original_candidate_line_sha256": "5ea24ed0d02357ed808bd425c88e3fd0999519efc31e2b6f9c820a0025ad9c3e" }, "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": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human 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-20261026-012-p68", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings 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": "c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c", "question_first_witness_sha256": "12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 9, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "aa86099942602f4fc1bcb4d40ec76aa6302a784c2857cba5a4767eefc77a4a0a", "arithmetic_steps": 6 }, "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." } }