{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261011-advance-20261015-r06-permutation-combination-0052", "pilot_number": "R0060-30877", "number": 30877, "family_id": "qf-density15-selection-one-distinguished-counts", "title": { "zh-Hant": "第 30877 題：含排固定物件的分割", "en": "Question 30877: Combinations Including or Excluding a Fixed Existing Label" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261011-advance-20261015-r06-permutation-combination-0052", "batch_id": "batch-20261011-advance-20261015-r06-permutation-combination-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-permutation-combination" ], "title_zh": "含排固定物件的分割", "stem_zh": "相異標記物件無放回，有序與無序分開。n=3≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。", "answer": { "kind": "expression", "value": "(1, 1, 0, 0)", "explanation_zh": "含排指定物件組合分割=(1, 1, 0, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "含排固定物件的分割" ] }, "variant": { "family_id": "qf-density15-selection-one-distinguished-counts", "seed": 202610156052, "parameters": { "n": 3, "k": 3, "model": "distinct_without_replacement", "domain": "finite_distinct_selection_protocol", "operation": "density15_selection_one_distinguished_counts", "semantic_template": "qf-density15-selection-one-distinguished-counts/[\"1\",\"1\",\"0\"]" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-11", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261011-advance-20261015-r06-permutation-combination-0052", "batch_id": "batch-20261011-advance-20261015-r06-permutation-combination-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "076f9a17168e50f9d6ed42a3f611e500703ac26ce2274b4a793e30233e30f1fa", "family_id": "qf-density15-selection-one-distinguished-counts", "seed": 202610156052, "semantic_template": "qf-density15-selection-one-distinguished-counts/[\"1\",\"1\",\"0\"]" }, "fields": { "title": "Combinations Including or Excluding a Fixed Existing Label", "stem": "Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=3≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).", "answer": { "kind": "expression", "value": "(1, 1, 0, 0)", "explanation": "Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (1, 1, 0, 0)." }, "skills": [ "Combinations Including or Excluding a Fixed Existing Label" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "相異標記物件無放回，有序與無序分開。n=3≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。", "en": "Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=3≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density15-selection-one-distinguished-counts, density15_selection_one_distinguished_counts, finite_distinct_selection_protocol]", "after": "derived = [[total, included_label0, excluded_label0, partition_defect]]", "relation": "implies", "rule_id": "R60_EXACT_perbank_operation_domain_FRAME_and_output_roles", "conditions": [], "explanation": { "zh-Hant": "density14/15用full-prefix underscores，density17用local hyphens。coordinate frame與circle frame不同，輸出欄序／型別依本family，不改原中英／JSON／status。", "en": "density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Selection all subsets ordered ORIGINAL available-choice factor0 := mul(1, 3)", "after": "Selection all subsets ordered ORIGINAL available-choice factor0 = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection all subsets ordered ORIGINAL available-choice factor0，不以小數近似替代。", "en": "Recompute Selection all subsets ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "3" ], "output": "3", "label": "Selection all subsets ordered ORIGINAL available-choice factor0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Selection all subsets ordered ORIGINAL available-choice factor1 := mul(3, 2)", "after": "Selection all subsets ordered ORIGINAL available-choice factor1 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection all subsets ordered ORIGINAL available-choice factor1，不以小數近似替代。", "en": "Recompute Selection all subsets ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "2" ], "output": "6", "label": "Selection all subsets ordered ORIGINAL available-choice factor1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Selection all subsets ordered ORIGINAL available-choice factor2 := mul(6, 1)", "after": "Selection all subsets ordered ORIGINAL available-choice factor2 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection all subsets ordered ORIGINAL available-choice factor2，不以小數近似替代。", "en": "Recompute Selection all subsets ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "1" ], "output": "6", "label": "Selection all subsets ordered ORIGINAL available-choice factor2" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [Selection all subsets ordered, 3, 3]", "after": "derived = [[3, 2, 1], 6]", "relation": "implies", "rule_id": "Selection_ORDERED_distinct_without_replacement_available_choices", "conditions": [], "explanation": { "zh-Hant": "第j位置有n-j個未用相異標記物件，完整乘k個choices；k0只有一個空ordered tuple。這不是可重複或環形／不可區分模型。", "en": "At position j there are n-j UNUSED distinct labeled objects.Multiply ALL k available-choice counts;at k0 there is exactly ONE empty ordered tuple.This is not replacement,circular or indistinguishable selection." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Selection all subsets within-subset ORIGINAL factorial factor2 := mul(1, 2)", "after": "Selection all subsets within-subset ORIGINAL factorial factor2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection all subsets within-subset ORIGINAL factorial factor2，不以小數近似替代。", "en": "Recompute Selection all subsets within-subset ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Selection all subsets within-subset ORIGINAL factorial factor2" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Selection all subsets within-subset ORIGINAL factorial factor3 := mul(2, 3)", "after": "Selection all subsets within-subset ORIGINAL factorial factor3 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection all subsets within-subset ORIGINAL factorial factor3，不以小數近似替代。", "en": "Recompute Selection all subsets within-subset ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Selection all subsets within-subset ORIGINAL factorial factor3" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [Selection all subsets within-subset, 3]", "after": "derived = [6]", "relation": "implies", "rule_id": "Selection_factorial_COMPLETE_integer_product_EMPTY0factorial1", "conditions": [], "explanation": { "zh-Hant": "完整乘2..n，n0或1是空積1；精確BigInt不作浮點、也不把衍生count截SAFE。", "en": "Multiply the COMPLETE2..n factors;at n0/1 the empty product is1.Use exact BigInt,not floats or a SAFE cap on derived counts." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Selection all subsets ORIGINAL exact ordered over k factorial := div(6, 6)", "after": "Selection all subsets ORIGINAL exact ordered over k factorial = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection all subsets ORIGINAL exact ordered over k factorial，不以小數近似替代。", "en": "Recompute Selection all subsets ORIGINAL exact ordered over k factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "6", "6" ], "output": "1", "label": "Selection all subsets ORIGINAL exact ordered over k factorial" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Selection all subsets reverse unordered times k factorial := mul(1, 6)", "after": "Selection all subsets reverse unordered times k factorial = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection all subsets reverse unordered times k factorial，不以小數近似替代。", "en": "Recompute Selection all subsets reverse unordered times k factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "6" ], "output": "6", "label": "Selection all subsets reverse unordered times k factorial" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [Selection all subsets, 3, 3, 6, 6]", "after": "derived = [1, 6]", "relation": "implies", "rule_id": "Selection_UNORDERED_each_subset_has_kfactorial_orderings", "conditions": [], "explanation": { "zh-Hant": "每個k-subset有恰k!個orderings，所以P/k!整除為C；反乘回P驗證。空subset與空ordering各一種，不自行改成概率模型。", "en": "EACH k-subset has exactly k! orderings,so P/k! is the exact integer C.Multiply back to verify P.An empty subset and empty ordering each count once;no probability model is asserted." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 2)", "after": "Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0，不以小數近似替代。", "en": "Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(2, 1)", "after": "Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1，不以小數近似替代。", "en": "Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "source roles = [Selection subsets INCLUDING label0 ordered, 2, 2]", "after": "derived = [[2, 1], 2]", "relation": "implies", "rule_id": "Selection_ORDERED_distinct_without_replacement_available_choices", "conditions": [], "explanation": { "zh-Hant": "第j位置有n-j個未用相異標記物件，完整乘k個choices；k0只有一個空ordered tuple。這不是可重複或環形／不可區分模型。", "en": "At position j there are n-j UNUSED distinct labeled objects.Multiply ALL k available-choice counts;at k0 there is exactly ONE empty ordered tuple.This is not replacement,circular or indistinguishable selection." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor2 := mul(1, 2)", "after": "Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor2，不以小數近似替代。", "en": "Recompute Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor2" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "source roles = [Selection subsets INCLUDING label0 within-subset, 2]", "after": "derived = [2]", "relation": "implies", "rule_id": "Selection_factorial_COMPLETE_integer_product_EMPTY0factorial1", "conditions": [], "explanation": { "zh-Hant": "完整乘2..n，n0或1是空積1；精確BigInt不作浮點、也不把衍生count截SAFE。", "en": "Multiply the COMPLETE2..n factors;at n0/1 the empty product is1.Use exact BigInt,not floats or a SAFE cap on derived counts." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial := div(2, 2)", "after": "Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial，不以小數近似替代。", "en": "Recompute Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2", "2" ], "output": "1", "label": "Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Selection subsets INCLUDING label0 reverse unordered times k factorial := mul(1, 2)", "after": "Selection subsets INCLUDING label0 reverse unordered times k factorial = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection subsets INCLUDING label0 reverse unordered times k factorial，不以小數近似替代。", "en": "Recompute Selection subsets INCLUDING label0 reverse unordered times k factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Selection subsets INCLUDING label0 reverse unordered times k factorial" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "source roles = [Selection subsets INCLUDING label0, 2, 2, 2, 2]", "after": "derived = [1, 2]", "relation": "implies", "rule_id": "Selection_UNORDERED_each_subset_has_kfactorial_orderings", "conditions": [], "explanation": { "zh-Hant": "每個k-subset有恰k!個orderings，所以P/k!整除為C；反乘回P驗證。空subset與空ordering各一種，不自行改成概率模型。", "en": "EACH k-subset has exactly k! orderings,so P/k! is the exact integer C.Multiply back to verify P.An empty subset and empty ordering each count once;no probability model is asserted." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "source roles = [Selection subsets EXCLUDING label0, 2, 3]", "after": "derived = [0]", "relation": "implies", "rule_id": "Selection_DERIVED_binomial_outside_size_ZERO_not_input_rejection", "conditions": [], "explanation": { "zh-Hant": "這是分割公式衍生的subset大小，不存在該大小的subset所以count0；不把k-1=-1誤作原輸入非法。", "en": "This is a DERIVED subset size in the partition formula.No subsets of that size exist,so the count is0;do not misclassify derived k-1=-1 as an invalid raw input." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Selection ORIGINAL included plus excluded := sum(1, 0)", "after": "Selection ORIGINAL included plus excluded = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection ORIGINAL included plus excluded，不以小數近似替代。", "en": "Recompute Selection ORIGINAL included plus excluded with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "0" ], "output": "1", "label": "Selection ORIGINAL included plus excluded" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Selection ORIGINAL partition sum minus whole := sub(1, 1)", "after": "Selection ORIGINAL partition sum minus whole = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Selection ORIGINAL partition sum minus whole，不以小數近似替代。", "en": "Recompute Selection ORIGINAL partition sum minus whole with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "1" ], "output": "0", "label": "Selection ORIGINAL partition sum minus whole" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "source roles = [3, 3]", "after": "derived = [1, 1, 0, 0]", "relation": "implies", "rule_id": "Selection_FIXED_EXISTING_label0_disjoint_complete_subset_partition", "conditions": [], "explanation": { "zh-Hant": "n>=1才有既定label0；包含它先移去0後選k-1，不含它只從其他n-1選k。兩類互斥且全覆蓋，k0包含0類count0、k=n排除類count0都合法。", "en": "n>=1 ensures the GIVEN existing label0.Containing it leaves k-1 choices from the other n-1 objects;excluding it chooses k from those n-1.The two classes are disjoint and exhaustive.At k0 inclusion count0,and at k=n exclusion count0 are legal." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": [ "1", "1", "0", "0" ] }, "conclusion": { "canonical_value": "(1, 1, 0, 0)", "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-20261011-advance-20261015-r06-permutation-combination-foundations", "batch_path": "public/data/questions/batches/batch-20261011-advance-20261015-r06-permutation-combination-foundations/questions.jsonl", "batch_sha256": "888e4e2c1b6b48f809197ef7f40084903294b320e8a43649b9768c6069b5ffb6", "original_line": 52, "original_line_sha256": "a27e5374140b1082b66a25d110081d0090b6253d19240ef87c4756e509d75b07", "question_compact_sha256": "076f9a17168e50f9d6ed42a3f611e500703ac26ce2274b4a793e30233e30f1fa", "english_row_compact_sha256": "f8c6f94e1b14a9d7e2adb12a962d1cdb42984dcb5a4f6adc48a6058fbf05d5e4", "english_manifest_path": "public/data/i18n/editions/batch-20261011-advance-20261015-r06-permutation-combination-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261011-advance-20261015-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": "322533190c1eb10144c4c59b6c70cc2f8105108b5f4f3a39dd577712e7185c02", "original_candidate_sha256": "0d6cbfef38408a98199b3dfb530f29d507e7af87f1308be049762d9f0e2091f2", "original_candidate_line_sha256": "0d6cbfef38408a98199b3dfb530f29d507e7af87f1308be049762d9f0e2091f2" }, "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-20261026-011-p61", "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": "d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf", "question_first_witness_sha256": "f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 23, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "a489a591ea0b5f898c2c117e9ac4835a037a88557fb07f5412c2a09a659c0c2c", "arithmetic_steps": 14 }, "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." } }