{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261011-advance-20261015-r06-permutation-combination-0038", "pilot_number": "R0060-30838", "number": 30838, "family_id": "qf-density15-selection-fixed-label-qualification", "title": { "zh-Hant": "第 30838 題：三個選取候選的資格軸", "en": "Question 30838: Candidate Selection Length, Typing, and Distinctness" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261011-advance-20261015-r06-permutation-combination-0038", "batch_id": "batch-20261011-advance-20261015-r06-permutation-combination-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-permutation-combination" ], "title_zh": "三個選取候選的資格軸", "stem_zh": "相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0], [0, 0], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。", "answer": { "kind": "expression", "value": "([否, 是, 是], [是, 是, 是], [是, 否, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])", "explanation_zh": "三個標記選取候選資格=([否, 是, 是], [是, 是, 是], [是, 否, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。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": [ "三個選取候選的資格軸" ] }, "variant": { "family_id": "qf-density15-selection-fixed-label-qualification", "seed": 202610156038, "parameters": { "n": 8, "k": 2, "candidates": [ [ 0 ], [ 0, 0 ], [ 0, 1 ] ], "model": "distinct_without_replacement", "domain": "finite_distinct_selection_protocol", "operation": "density15_selection_fixed_label_qualification", "semantic_template": "qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[true,true,false,\"invalid\"],[true,true,true,[0,1]]]" } }, "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-0038", "batch_id": "batch-20261011-advance-20261015-r06-permutation-combination-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "fe7ceaab207293c0ffd7f22d0cfc82a3870f3834f8a40bf979846d760cfcee5b", "family_id": "qf-density15-selection-fixed-label-qualification", "seed": 202610156038, "semantic_template": "qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[true,true,false,\"invalid\"],[true,true,true,[0,1]]]" }, "fields": { "title": "Candidate Selection Length, Typing, and Distinctness", "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. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0, 0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.", "answer": { "kind": "expression", "value": "([No, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])", "explanation": "Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])." }, "skills": [ "Candidate Selection Length, Typing, and Distinctness" ], "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。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0], [0, 0], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。", "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. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0, 0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density15-selection-fixed-label-qualification, density15_selection_fixed_label_qualification, finite_distinct_selection_protocol]", "after": "derived = [[length_flags, typed_flags, distinct_flags, full_flags, ordered_tuples_or_invalid, sorted_subsets_or_invalid]]", "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": "source roles = [8, 2, 0, [0]]", "after": "derived = [否, 是, 是, 否, 不合選取, 不合選取]", "relation": "implies", "rule_id": "Selection_EACH_candidate_LENGTH_TYPE_DISTINCT_axes_before_full_guard", "conditions": [], "explanation": { "zh-Hant": "raw候選可錯長度、超域或重複label，先分三條件再合取，不能先reject掉合法No案例。只對good保原ordered row並另列sorted subset；k0空row在Fin0也合法，三個原位置保留。", "en": "Wrong-length,outside-domain or repeated-label RAW candidates are legal No cases.Check LENGTH,TYPE andDISTINCTNESS separately,then conjoin.ONLY a good candidate yields its ORIGINAL ordered row and separately sorted subset.At k0 an empty row is valid even in Fin0;retain all three candidate positions." }, "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": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [8, 2, 1, [0, 0]]", "after": "derived = [是, 是, 否, 否, 不合選取, 不合選取]", "relation": "implies", "rule_id": "Selection_EACH_candidate_LENGTH_TYPE_DISTINCT_axes_before_full_guard", "conditions": [], "explanation": { "zh-Hant": "raw候選可錯長度、超域或重複label，先分三條件再合取，不能先reject掉合法No案例。只對good保原ordered row並另列sorted subset；k0空row在Fin0也合法，三個原位置保留。", "en": "Wrong-length,outside-domain or repeated-label RAW candidates are legal No cases.Check LENGTH,TYPE andDISTINCTNESS separately,then conjoin.ONLY a good candidate yields its ORIGINAL ordered row and separately sorted subset.At k0 an empty row is valid even in Fin0;retain all three candidate positions." }, "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": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [8, 2, 2, [0, 1]]", "after": "derived = [是, 是, 是, 是, [0, 1], [0, 1]]", "relation": "implies", "rule_id": "Selection_EACH_candidate_LENGTH_TYPE_DISTINCT_axes_before_full_guard", "conditions": [], "explanation": { "zh-Hant": "raw候選可錯長度、超域或重複label，先分三條件再合取，不能先reject掉合法No案例。只對good保原ordered row並另列sorted subset；k0空row在Fin0也合法，三個原位置保留。", "en": "Wrong-length,outside-domain or repeated-label RAW candidates are legal No cases.Check LENGTH,TYPE andDISTINCTNESS separately,then conjoin.ONLY a good candidate yields its ORIGINAL ordered row and separately sorted subset.At k0 an empty row is valid even in Fin0;retain all three candidate positions." }, "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": [ [ false, true, true ], [ true, true, true ], [ true, false, true ], [ false, false, true ], [ "不合選取", "不合選取", [ 0, 1 ] ], [ "不合選取", "不合選取", [ 0, 1 ] ] ] }, "conclusion": { "canonical_value": "([否, 是, 是], [是, 是, 是], [是, 否, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])", "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": 38, "original_line_sha256": "f52c33a4ad73db46c248cf204a406d29af7b0a9a6c1cef1a45447d05cbe0664f", "question_compact_sha256": "fe7ceaab207293c0ffd7f22d0cfc82a3870f3834f8a40bf979846d760cfcee5b", "english_row_compact_sha256": "0d0c02abb1fb587a1288d41e96a19b7216ec5c409003e394c436e0061b9bb569", "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": "ff9d8b63c25b91f3ec2892a75819dbe5c34f8683b129b9198c24f4650ab9ba96", "original_candidate_line_sha256": "ff9d8b63c25b91f3ec2892a75819dbe5c34f8683b129b9198c24f4650ab9ba96" }, "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": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "6733a03235ed13246c984c57780ae054427d3839f366aeb1f76104a55e384056", "arithmetic_steps": 0 }, "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." } }