{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261011-advance-20261015-r06-permutation-combination-0092", "pilot_number": "R0060-30867", "number": 30867, "family_id": "qf-density15-selection-forced-initial-order", "title": { "zh-Hant": "第 30867 題：固定前序的完成計數", "en": "Question 30867: Completions after Fixing an Initial Ordered Prefix" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261011-advance-20261015-r06-permutation-combination-0092", "batch_id": "batch-20261011-advance-20261015-r06-permutation-combination-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-permutation-combination" ], "title_zh": "固定前序的完成計數", "stem_zh": "相異標記物件無放回，有序與無序分開。n=17、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。", "answer": { "kind": "expression", "value": "(16, 2, 240, 4080, 1/17)", "explanation_zh": "固定前序餘量與完成計數比=(16, 2, 240, 4080, 1/17)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。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": 2, "skills": [ "固定前序的完成計數" ] }, "variant": { "family_id": "qf-density15-selection-forced-initial-order", "seed": 202610156092, "parameters": { "n": 17, "k": 3, "r": 1, "model": "distinct_without_replacement", "domain": "finite_distinct_selection_protocol", "operation": "density15_selection_forced_initial_order", "semantic_template": "qf-density15-selection-forced-initial-order/[16,2,\"240\",\"4080\"]" } }, "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-0092", "batch_id": "batch-20261011-advance-20261015-r06-permutation-combination-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "1ca46d6acf9b60e49394d198a6434b49bfedc21464c46fcc0f516fc045ac3b5d", "family_id": "qf-density15-selection-forced-initial-order", "seed": 202610156092, "semantic_template": "qf-density15-selection-forced-initial-order/[16,2,\"240\",\"4080\"]" }, "fields": { "title": "Completions after Fixing an Initial Ordered Prefix", "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=17, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).", "answer": { "kind": "expression", "value": "(16, 2, 240, 4080, 1/17)", "explanation": "Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (16, 2, 240, 4080, 1/17)." }, "skills": [ "Completions after Fixing an Initial Ordered Prefix" ], "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=17、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/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=17, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density15-selection-forced-initial-order, density15_selection_forced_initial_order, finite_distinct_selection_protocol]", "after": "derived = [[remaining_objects, remaining_positions, completions, full_ordered_count, COUNT_ratio]]", "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 prefix ORIGINAL n minus used r := sub(17, 1)", "after": "Selection prefix ORIGINAL n minus used r = 16", "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 prefix ORIGINAL n minus used r，不以小數近似替代。", "en": "Recompute Selection prefix ORIGINAL n minus used r with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "17", "1" ], "output": "16", "label": "Selection prefix ORIGINAL n minus used r" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Selection prefix ORIGINAL k minus filled r := sub(3, 1)", "after": "Selection prefix ORIGINAL k minus filled r = 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 prefix ORIGINAL k minus filled r，不以小數近似替代。", "en": "Recompute Selection prefix ORIGINAL k minus filled r with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "3", "1" ], "output": "2", "label": "Selection prefix ORIGINAL k minus filled r" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Selection prefix remaining ordered ORIGINAL available-choice factor0 := mul(1, 16)", "after": "Selection prefix remaining ordered ORIGINAL available-choice factor0 = 16", "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 prefix remaining ordered ORIGINAL available-choice factor0，不以小數近似替代。", "en": "Recompute Selection prefix remaining ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "16" ], "output": "16", "label": "Selection prefix remaining ordered ORIGINAL available-choice factor0" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Selection prefix remaining ordered ORIGINAL available-choice factor1 := mul(16, 15)", "after": "Selection prefix remaining ordered ORIGINAL available-choice factor1 = 240", "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 prefix remaining ordered ORIGINAL available-choice factor1，不以小數近似替代。", "en": "Recompute Selection prefix remaining ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "16", "15" ], "output": "240", "label": "Selection prefix remaining ordered ORIGINAL available-choice factor1" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [Selection prefix remaining ordered, 16, 2]", "after": "derived = [[16, 15], 240]", "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": "S007", "uses": [ "G001", "S006" ], "before": "Selection full ordered ORIGINAL available-choice factor0 := mul(1, 17)", "after": "Selection full ordered ORIGINAL available-choice factor0 = 17", "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 full ordered ORIGINAL available-choice factor0，不以小數近似替代。", "en": "Recompute Selection full ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "17" ], "output": "17", "label": "Selection full ordered ORIGINAL available-choice factor0" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Selection full ordered ORIGINAL available-choice factor1 := mul(17, 16)", "after": "Selection full ordered ORIGINAL available-choice factor1 = 272", "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 full ordered ORIGINAL available-choice factor1，不以小數近似替代。", "en": "Recompute Selection full ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "17", "16" ], "output": "272", "label": "Selection full ordered ORIGINAL available-choice factor1" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Selection full ordered ORIGINAL available-choice factor2 := mul(272, 15)", "after": "Selection full ordered ORIGINAL available-choice factor2 = 4080", "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 full ordered ORIGINAL available-choice factor2，不以小數近似替代。", "en": "Recompute Selection full ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "272", "15" ], "output": "4080", "label": "Selection full ordered ORIGINAL available-choice factor2" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [Selection full ordered, 17, 3]", "after": "derived = [[17, 16, 15], 4080]", "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": "S011", "uses": [ "G001", "S010" ], "before": "Selection ORIGINAL completion COUNT ratio := div(240, 4080)", "after": "Selection ORIGINAL completion COUNT ratio = 1/17", "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 completion COUNT ratio，不以小數近似替代。", "en": "Recompute Selection ORIGINAL completion COUNT ratio with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "240", "4080" ], "output": "1/17", "label": "Selection ORIGINAL completion COUNT ratio" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [17, 3, 1, [0]]", "after": "derived = [16, 2, 240, 4080, 1/17]", "relation": "implies", "rule_id": "Selection_FIXED_prefix_labels_and_COUNT_ratio_NOT_probability", "conditions": [], "explanation": { "zh-Hant": "前r位置固定用0..r-1且次序已給，先扣已用物件／已填位置。r0不減count，r=k只一個completion；最後是count比，沒有uniform law不可自行叫概率。", "en": "The FIRST r positions use labels0..r-1 in their GIVEN order.Subtract used objects/filled positions first.At r0 retain the full count;at r=k there is one completion.The final value is a COUNT ratio,not a probability without a uniform law." }, "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": [ 16, 2, "240", "4080", "1/17" ] }, "conclusion": { "canonical_value": "(16, 2, 240, 4080, 1/17)", "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": 92, "original_line_sha256": "904d7cbb371eff341e3ff98d11787bb557c4c96fb4c8692f9d1bf9053b422467", "question_compact_sha256": "1ca46d6acf9b60e49394d198a6434b49bfedc21464c46fcc0f516fc045ac3b5d", "english_row_compact_sha256": "987f5c7496fbce8cba461b78392cbca607bfc20c207b624218c3b910c114209d", "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": "5f9a707abbc150f4c55f11559c1296f7e40b0eb678c9b5e9b1d1788411e4aae9", "original_candidate_line_sha256": "5f9a707abbc150f4c55f11559c1296f7e40b0eb678c9b5e9b1d1788411e4aae9" }, "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": 12, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "e205260cee06b756ee93a0761641cc3b24dbb62cde61880b9c40a12b144677f4", "arithmetic_steps": 8 }, "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." } }