{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260930-r06-combination-exact-count-0004",
  "pilot_number": "R0005-03004",
  "number": 3004,
  "family_id": "qf-combination-exact-count",
  "title": {
    "zh-Hant": "第 3004 題：無序不重複選取",
    "en": "Question 3004: Exact Unordered Selection Count"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260930-r06-combination-exact-count-0004",
    "batch_id": "batch-20260930-r06-permutation-combination-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-permutation-combination"
    ],
    "title_zh": "無序不重複選取",
    "stem_zh": "從 2 個彼此相異物件無放回且不重複選取 2 個，空選取計一種。忽略順序，求無序選取數 C(2,2)。",
    "answer": {
      "kind": "short_text",
      "value": "1",
      "explanation_zh": "按是否包含新加入物件的子集合分類遞迴計數，得到 1；空子集合恰有一個。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "相異物件",
        "無序選取"
      ]
    },
    "variant": {
      "family_id": "qf-combination-exact-count",
      "seed": 202609306104,
      "parameters": {
        "operation": "combination",
        "n": 2,
        "k": 2,
        "semantic_template": "qf-combination-exact-count/{\"k\":2,\"n\":2,\"operation\":\"combination\"}"
      }
    },
    "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-combination-exact-count-0004",
    "batch_id": "batch-20260930-r06-permutation-combination-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "78df56b824ce975b03272dbf42f60792d09d14894babd7bc49573059d821a4de",
      "family_id": "qf-combination-exact-count",
      "seed": 202609306104,
      "semantic_template": "qf-combination-exact-count/{\"k\":2,\"n\":2,\"operation\":\"combination\"}"
    },
    "fields": {
      "title": "Exact Unordered Selection Count",
      "stem": "Select k=2 objects without order, replacement, or repetition from n=2 distinct objects. The empty selection counts once. Find C(2,2).",
      "answer": {
        "kind": "short_text",
        "value": "1",
        "explanation": "Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection."
      },
      "skills": [
        "distinct objects",
        "unordered 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 2 個彼此相異物件無放回且不重複選取 2 個，空選取計一種。忽略順序，求無序選取數 C(2,2)。",
        "en": "Select k=2 objects without order, replacement, or repetition from n=2 distinct objects. The empty selection counts once. Find C(2,2)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "n=2;k=2;distinct objects,without replacement",
      "after": "unordered k-subsets of an n-element set",
      "relation": "implies",
      "rule_id": "unordered_selection_model",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個物件至多取一次，忽略先後順序；排列數不能直接當作本題答案。",
        "en": "Each object is selected at most once and order is ignored;the ordered-selection count is not the requested answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "choices at position1 := sub(2, 0)",
      "after": "choices at position1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已選i個互異物件後，剩n−i個可選。",
        "en": "After i distinct selections,n−i objects remain available."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 choices at position1，不以小數近似替代。",
          "en": "Recompute choices at position1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "choices at position1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "ordered prefix count1 := mul(1, 2)",
      "after": "ordered prefix count1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "乘法原理累積有序選取數，每個位置仍不重複。",
        "en": "The product rule accumulates the ordered count without repetition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ordered prefix count1，不以小數近似替代。",
          "en": "Recompute ordered prefix count1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "ordered prefix count1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "choices at position2 := sub(2, 1)",
      "after": "choices at position2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已選i個互異物件後，剩n−i個可選。",
        "en": "After i distinct selections,n−i objects remain available."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 choices at position2，不以小數近似替代。",
          "en": "Recompute choices at position2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "1"
        ],
        "output": "1",
        "label": "choices at position2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "ordered prefix count2 := mul(2, 1)",
      "after": "ordered prefix count2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "乘法原理累積有序選取數，每個位置仍不重複。",
        "en": "The product rule accumulates the ordered count without repetition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ordered prefix count2，不以小數近似替代。",
          "en": "Recompute ordered prefix count2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "ordered prefix count2"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "factorial through2 := mul(1, 2)",
      "after": "factorial through2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "累乘求k!，即同一k元素集合的全部排列數。",
        "en": "Accumulate k!,the number of orders of the same selected k-element set."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 factorial through2，不以小數近似替代。",
          "en": "Recompute factorial through2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "factorial through2"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "ordered selections=2;orders per subset=2",
      "after": "C(2,2)=ordered selections/k!",
      "relation": "implies",
      "rule_id": "uniform_order_fibers",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個k子集合恰對應k!個排列，且不同子集合的排列互不重疊；因此除以相同的正整數k!。",
        "en": "Every k-subset has exactly k! orderings,and different subsets have disjoint orderings;divide by this common positive factor."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "unordered subset count := div(2, 2)",
      "after": "unordered subset count = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "精確相除得到整數；k=0時兩個空乘積均為1，空選取恰有一種。",
        "en": "Exact division gives an integer;when k=0 both empty products are1 and there is exactly one empty selection."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 unordered subset count，不以小數近似替代。",
          "en": "Recompute unordered subset count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "2"
        ],
        "output": "1",
        "label": "unordered subset count"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "C(2,2)=n!/(k!·(n−k)!)",
      "after": "C(2,2)=2/(2·1)=1",
      "relation": "implies",
      "rule_id": "binomial_count_formula",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把下降乘積改寫為n!/(n−k)!，再核對組合公式及所有邊界值。",
        "en": "Rewrite the falling product as n!/(n−k)! and audit the binomial formula,including boundary cases."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "binomial_count",
        "inputs": [
          "2",
          "2"
        ],
        "output": "1",
        "label": "combination"
      }
    }
  ],
  "math_model": {
    "result": "1",
    "ordered_count": "2",
    "orders_per_subset": "2"
  },
  "conclusion": {
    "canonical_value": "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": {
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    "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",
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      "original_line_sha256": "5eb935087c9583ca20b1e89c0bd3ad62416a0f50697bee0e0a8224a44a08fefe",
      "question_compact_sha256": "78df56b824ce975b03272dbf42f60792d09d14894babd7bc49573059d821a4de",
      "english_row_compact_sha256": "ce9a5e75b4a30fb62a7daad8ff325dd1055681dddce76703f19a736345d6102b",
      "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": {
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        "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": "66b95c054a23815e6aa9999f9605667abcb96e0c5a97b71cd67b67041065878a",
    "original_candidate_sha256": "68ed55a3118846cf81fc30eaae2e94f16cd19d5a3b50e9cbeda6898b911adcc9",
    "original_candidate_line_sha256": "68ed55a3118846cf81fc30eaae2e94f16cd19d5a3b50e9cbeda6898b911adcc9"
  },
  "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-20261026-004-p6",
    "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": "5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2",
      "question_first_witness_sha256": "1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9"
    },
    "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": "ac71ab7c89f25f7114ba35b7956fe26ef6b5a9d10544a2c2d353d5977454f255",
      "arithmetic_steps": 7
    },
    "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."
  }
}