{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0001","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 0 個彼此相異物件無放回且不重複選取 0 個，空選取計一種。順序有意義，求有序選取數 P(0,0)。","answer":{"kind":"short_text","value":"1","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 1；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306001,"parameters":{"operation":"permutation","n":0,"k":0,"semantic_template":"qf-permutation-exact-count/{\"k\":0,\"n\":0,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0002","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 1 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。順序有意義，求有序選取數 P(1,1)。","answer":{"kind":"short_text","value":"1","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 1；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306002,"parameters":{"operation":"permutation","n":1,"k":1,"semantic_template":"qf-permutation-exact-count/{\"k\":1,\"n\":1,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0003","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 2 個彼此相異物件無放回且不重複選取 0 個，空選取計一種。順序有意義，求有序選取數 P(2,0)。","answer":{"kind":"short_text","value":"1","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 1；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306003,"parameters":{"operation":"permutation","n":2,"k":0,"semantic_template":"qf-permutation-exact-count/{\"k\":0,\"n\":2,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0004","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","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。順序有意義，求有序選取數 P(2,2)。","answer":{"kind":"short_text","value":"2","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 2；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306004,"parameters":{"operation":"permutation","n":2,"k":2,"semantic_template":"qf-permutation-exact-count/{\"k\":2,\"n\":2,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0005","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 3 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。順序有意義，求有序選取數 P(3,1)。","answer":{"kind":"short_text","value":"3","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 3；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306005,"parameters":{"operation":"permutation","n":3,"k":1,"semantic_template":"qf-permutation-exact-count/{\"k\":1,\"n\":3,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0006","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 3 個彼此相異物件無放回且不重複選取 3 個，空選取計一種。順序有意義，求有序選取數 P(3,3)。","answer":{"kind":"short_text","value":"6","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 6；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306006,"parameters":{"operation":"permutation","n":3,"k":3,"semantic_template":"qf-permutation-exact-count/{\"k\":3,\"n\":3,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0007","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","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。順序有意義，求有序選取數 P(4,2)。","answer":{"kind":"short_text","value":"12","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 12；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306007,"parameters":{"operation":"permutation","n":4,"k":2,"semantic_template":"qf-permutation-exact-count/{\"k\":2,\"n\":4,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0008","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 4 個彼此相異物件無放回且不重複選取 4 個，空選取計一種。順序有意義，求有序選取數 P(4,4)。","answer":{"kind":"short_text","value":"24","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 24；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306008,"parameters":{"operation":"permutation","n":4,"k":4,"semantic_template":"qf-permutation-exact-count/{\"k\":4,\"n\":4,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0009","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 5 個彼此相異物件無放回且不重複選取 0 個，空選取計一種。順序有意義，求有序選取數 P(5,0)。","answer":{"kind":"short_text","value":"1","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 1；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306009,"parameters":{"operation":"permutation","n":5,"k":0,"semantic_template":"qf-permutation-exact-count/{\"k\":0,\"n\":5,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0010","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 5 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。順序有意義，求有序選取數 P(5,1)。","answer":{"kind":"short_text","value":"5","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 5；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306010,"parameters":{"operation":"permutation","n":5,"k":1,"semantic_template":"qf-permutation-exact-count/{\"k\":1,\"n\":5,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0011","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 5 個彼此相異物件無放回且不重複選取 2 個，空選取計一種。順序有意義，求有序選取數 P(5,2)。","answer":{"kind":"short_text","value":"20","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 20；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306011,"parameters":{"operation":"permutation","n":5,"k":2,"semantic_template":"qf-permutation-exact-count/{\"k\":2,\"n\":5,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0012","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 5 個彼此相異物件無放回且不重複選取 5 個，空選取計一種。順序有意義，求有序選取數 P(5,5)。","answer":{"kind":"short_text","value":"120","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 120；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306012,"parameters":{"operation":"permutation","n":5,"k":5,"semantic_template":"qf-permutation-exact-count/{\"k\":5,\"n\":5,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0013","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 6 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。順序有意義，求有序選取數 P(6,1)。","answer":{"kind":"short_text","value":"6","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 6；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306013,"parameters":{"operation":"permutation","n":6,"k":1,"semantic_template":"qf-permutation-exact-count/{\"k\":1,\"n\":6,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0014","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 6 個彼此相異物件無放回且不重複選取 3 個，空選取計一種。順序有意義，求有序選取數 P(6,3)。","answer":{"kind":"short_text","value":"120","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 120；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306014,"parameters":{"operation":"permutation","n":6,"k":3,"semantic_template":"qf-permutation-exact-count/{\"k\":3,\"n\":6,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0015","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 6 個彼此相異物件無放回且不重複選取 6 個，空選取計一種。順序有意義，求有序選取數 P(6,6)。","answer":{"kind":"short_text","value":"720","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 720；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306015,"parameters":{"operation":"permutation","n":6,"k":6,"semantic_template":"qf-permutation-exact-count/{\"k\":6,\"n\":6,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0016","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 7 個彼此相異物件無放回且不重複選取 2 個，空選取計一種。順序有意義，求有序選取數 P(7,2)。","answer":{"kind":"short_text","value":"42","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 42；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306016,"parameters":{"operation":"permutation","n":7,"k":2,"semantic_template":"qf-permutation-exact-count/{\"k\":2,\"n\":7,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0017","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 7 個彼此相異物件無放回且不重複選取 4 個，空選取計一種。順序有意義，求有序選取數 P(7,4)。","answer":{"kind":"short_text","value":"840","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 840；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306017,"parameters":{"operation":"permutation","n":7,"k":4,"semantic_template":"qf-permutation-exact-count/{\"k\":4,\"n\":7,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0018","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 7 個彼此相異物件無放回且不重複選取 7 個，空選取計一種。順序有意義，求有序選取數 P(7,7)。","answer":{"kind":"short_text","value":"5040","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 5040；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306018,"parameters":{"operation":"permutation","n":7,"k":7,"semantic_template":"qf-permutation-exact-count/{\"k\":7,\"n\":7,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0019","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 8 個彼此相異物件無放回且不重複選取 0 個，空選取計一種。順序有意義，求有序選取數 P(8,0)。","answer":{"kind":"short_text","value":"1","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 1；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306019,"parameters":{"operation":"permutation","n":8,"k":0,"semantic_template":"qf-permutation-exact-count/{\"k\":0,\"n\":8,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0020","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 8 個彼此相異物件無放回且不重複選取 3 個，空選取計一種。順序有意義，求有序選取數 P(8,3)。","answer":{"kind":"short_text","value":"336","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 336；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306020,"parameters":{"operation":"permutation","n":8,"k":3,"semantic_template":"qf-permutation-exact-count/{\"k\":3,\"n\":8,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0021","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 8 個彼此相異物件無放回且不重複選取 8 個，空選取計一種。順序有意義，求有序選取數 P(8,8)。","answer":{"kind":"short_text","value":"40320","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 40320；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306021,"parameters":{"operation":"permutation","n":8,"k":8,"semantic_template":"qf-permutation-exact-count/{\"k\":8,\"n\":8,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0022","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 9 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。順序有意義，求有序選取數 P(9,1)。","answer":{"kind":"short_text","value":"9","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 9；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306022,"parameters":{"operation":"permutation","n":9,"k":1,"semantic_template":"qf-permutation-exact-count/{\"k\":1,\"n\":9,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0023","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 9 個彼此相異物件無放回且不重複選取 4 個，空選取計一種。順序有意義，求有序選取數 P(9,4)。","answer":{"kind":"short_text","value":"3024","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 3024；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306023,"parameters":{"operation":"permutation","n":9,"k":4,"semantic_template":"qf-permutation-exact-count/{\"k\":4,\"n\":9,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0024","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 10 個彼此相異物件無放回且不重複選取 5 個，空選取計一種。順序有意義，求有序選取數 P(10,5)。","answer":{"kind":"short_text","value":"30240","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 30240；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306024,"parameters":{"operation":"permutation","n":10,"k":5,"semantic_template":"qf-permutation-exact-count/{\"k\":5,\"n\":10,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-exact-count-0025","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序不重複選取","stem_zh":"從 12 個彼此相異物件無放回且不重複選取 6 個，空選取計一種。順序有意義，求有序選取數 P(12,6)。","answer":{"kind":"short_text","value":"665280","explanation_zh":"依序相乘每一步尚可選的數量，精確計數為 665280；k=0 時使用空乘積 1。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相異物件","有序選取"]},"variant":{"family_id":"qf-permutation-exact-count","seed":202609306025,"parameters":{"operation":"permutation","n":12,"k":6,"semantic_template":"qf-permutation-exact-count/{\"k\":6,\"n\":12,\"operation\":\"permutation\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0001","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 0 個彼此相異物件無放回且不重複選取 0 個，空選取計一種。忽略順序，求無序選取數 C(0,0)。","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":202609306101,"parameters":{"operation":"combination","n":0,"k":0,"semantic_template":"qf-combination-exact-count/{\"k\":0,\"n\":0,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0002","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 1 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。忽略順序，求無序選取數 C(1,1)。","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":2,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306102,"parameters":{"operation":"combination","n":1,"k":1,"semantic_template":"qf-combination-exact-count/{\"k\":1,\"n\":1,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0003","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 0 個，空選取計一種。忽略順序，求無序選取數 C(2,0)。","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":3,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306103,"parameters":{"operation":"combination","n":2,"k":0,"semantic_template":"qf-combination-exact-count/{\"k\":0,\"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"}}
{"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0005","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 3 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。忽略順序，求無序選取數 C(3,1)。","answer":{"kind":"short_text","value":"3","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 3；空子集合恰有一個。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306105,"parameters":{"operation":"combination","n":3,"k":1,"semantic_template":"qf-combination-exact-count/{\"k\":1,\"n\":3,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0006","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 3 個彼此相異物件無放回且不重複選取 3 個，空選取計一種。忽略順序，求無序選取數 C(3,3)。","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":3,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306106,"parameters":{"operation":"combination","n":3,"k":3,"semantic_template":"qf-combination-exact-count/{\"k\":3,\"n\":3,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0007","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 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。忽略順序，求無序選取數 C(4,2)。","answer":{"kind":"short_text","value":"6","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 6；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":202609306107,"parameters":{"operation":"combination","n":4,"k":2,"semantic_template":"qf-combination-exact-count/{\"k\":2,\"n\":4,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0008","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 4 個彼此相異物件無放回且不重複選取 4 個，空選取計一種。忽略順序，求無序選取數 C(4,4)。","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":2,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306108,"parameters":{"operation":"combination","n":4,"k":4,"semantic_template":"qf-combination-exact-count/{\"k\":4,\"n\":4,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0009","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 5 個彼此相異物件無放回且不重複選取 0 個，空選取計一種。忽略順序，求無序選取數 C(5,0)。","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":3,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306109,"parameters":{"operation":"combination","n":5,"k":0,"semantic_template":"qf-combination-exact-count/{\"k\":0,\"n\":5,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0010","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 5 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。忽略順序，求無序選取數 C(5,1)。","answer":{"kind":"short_text","value":"5","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 5；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":202609306110,"parameters":{"operation":"combination","n":5,"k":1,"semantic_template":"qf-combination-exact-count/{\"k\":1,\"n\":5,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0011","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 5 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(5,2)。","answer":{"kind":"short_text","value":"10","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 10；空子集合恰有一個。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306111,"parameters":{"operation":"combination","n":5,"k":2,"semantic_template":"qf-combination-exact-count/{\"k\":2,\"n\":5,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0012","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 5 個彼此相異物件無放回且不重複選取 5 個，空選取計一種。忽略順序，求無序選取數 C(5,5)。","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":3,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306112,"parameters":{"operation":"combination","n":5,"k":5,"semantic_template":"qf-combination-exact-count/{\"k\":5,\"n\":5,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0013","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 6 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。忽略順序，求無序選取數 C(6,1)。","answer":{"kind":"short_text","value":"6","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 6；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":202609306113,"parameters":{"operation":"combination","n":6,"k":1,"semantic_template":"qf-combination-exact-count/{\"k\":1,\"n\":6,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0014","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 6 個彼此相異物件無放回且不重複選取 3 個，空選取計一種。忽略順序，求無序選取數 C(6,3)。","answer":{"kind":"short_text","value":"20","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 20；空子集合恰有一個。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306114,"parameters":{"operation":"combination","n":6,"k":3,"semantic_template":"qf-combination-exact-count/{\"k\":3,\"n\":6,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0015","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 6 個彼此相異物件無放回且不重複選取 6 個，空選取計一種。忽略順序，求無序選取數 C(6,6)。","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":3,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306115,"parameters":{"operation":"combination","n":6,"k":6,"semantic_template":"qf-combination-exact-count/{\"k\":6,\"n\":6,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0016","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 7 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(7,2)。","answer":{"kind":"short_text","value":"21","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 21；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":202609306116,"parameters":{"operation":"combination","n":7,"k":2,"semantic_template":"qf-combination-exact-count/{\"k\":2,\"n\":7,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0017","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 7 個彼此相異物件無放回且不重複選取 4 個，空選取計一種。忽略順序，求無序選取數 C(7,4)。","answer":{"kind":"short_text","value":"35","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 35；空子集合恰有一個。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306117,"parameters":{"operation":"combination","n":7,"k":4,"semantic_template":"qf-combination-exact-count/{\"k\":4,\"n\":7,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0018","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 7 個彼此相異物件無放回且不重複選取 7 個，空選取計一種。忽略順序，求無序選取數 C(7,7)。","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":3,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306118,"parameters":{"operation":"combination","n":7,"k":7,"semantic_template":"qf-combination-exact-count/{\"k\":7,\"n\":7,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0019","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 8 個彼此相異物件無放回且不重複選取 0 個，空選取計一種。忽略順序，求無序選取數 C(8,0)。","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":202609306119,"parameters":{"operation":"combination","n":8,"k":0,"semantic_template":"qf-combination-exact-count/{\"k\":0,\"n\":8,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0020","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 8 個彼此相異物件無放回且不重複選取 3 個，空選取計一種。忽略順序，求無序選取數 C(8,3)。","answer":{"kind":"short_text","value":"56","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 56；空子集合恰有一個。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306120,"parameters":{"operation":"combination","n":8,"k":3,"semantic_template":"qf-combination-exact-count/{\"k\":3,\"n\":8,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0021","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 8 個彼此相異物件無放回且不重複選取 8 個，空選取計一種。忽略順序，求無序選取數 C(8,8)。","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":3,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306121,"parameters":{"operation":"combination","n":8,"k":8,"semantic_template":"qf-combination-exact-count/{\"k\":8,\"n\":8,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0022","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 9 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。忽略順序，求無序選取數 C(9,1)。","answer":{"kind":"short_text","value":"9","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 9；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":202609306122,"parameters":{"operation":"combination","n":9,"k":1,"semantic_template":"qf-combination-exact-count/{\"k\":1,\"n\":9,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0023","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 9 個彼此相異物件無放回且不重複選取 4 個，空選取計一種。忽略順序，求無序選取數 C(9,4)。","answer":{"kind":"short_text","value":"126","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 126；空子集合恰有一個。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306123,"parameters":{"operation":"combination","n":9,"k":4,"semantic_template":"qf-combination-exact-count/{\"k\":4,\"n\":9,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0024","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 10 個彼此相異物件無放回且不重複選取 5 個，空選取計一種。忽略順序，求無序選取數 C(10,5)。","answer":{"kind":"short_text","value":"252","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 252；空子集合恰有一個。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相異物件","無序選取"]},"variant":{"family_id":"qf-combination-exact-count","seed":202609306124,"parameters":{"operation":"combination","n":10,"k":5,"semantic_template":"qf-combination-exact-count/{\"k\":5,\"n\":10,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-combination-exact-count-0025","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 12 個彼此相異物件無放回且不重複選取 6 個，空選取計一種。忽略順序，求無序選取數 C(12,6)。","answer":{"kind":"short_text","value":"924","explanation_zh":"按是否包含新加入物件的子集合分類遞迴計數，得到 924；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":202609306125,"parameters":{"operation":"combination","n":12,"k":6,"semantic_template":"qf-combination-exact-count/{\"k\":6,\"n\":12,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0001","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 0 個彼此相異物件無放回且不重複選取 0 個，空選取計一種。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306201,"parameters":{"operation":"model_choice","n":0,"k":0,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":0,\"n\":0,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0002","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 1 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。本題忽略選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"組合","explanation_zh":"相異性、無放回及不重複均已明示；順序被忽略，所以選擇組合。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306202,"parameters":{"operation":"model_choice","n":1,"k":1,"ordered":false,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":1,\"n\":1,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0003","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 2 個彼此相異物件無放回且不重複選取 0 個，空選取計一種。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306203,"parameters":{"operation":"model_choice","n":2,"k":0,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":0,\"n\":2,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0004","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","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。本題忽略選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"組合","explanation_zh":"相異性、無放回及不重複均已明示；順序被忽略，所以選擇組合。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306204,"parameters":{"operation":"model_choice","n":2,"k":2,"ordered":false,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":2,\"n\":2,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0005","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 3 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306205,"parameters":{"operation":"model_choice","n":3,"k":1,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":1,\"n\":3,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0006","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 3 個彼此相異物件無放回且不重複選取 3 個，空選取計一種。本題忽略選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"組合","explanation_zh":"相異性、無放回及不重複均已明示；順序被忽略，所以選擇組合。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306206,"parameters":{"operation":"model_choice","n":3,"k":3,"ordered":false,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":3,\"n\":3,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0007","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","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。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306207,"parameters":{"operation":"model_choice","n":4,"k":2,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":2,\"n\":4,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0008","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 4 個彼此相異物件無放回且不重複選取 4 個，空選取計一種。本題忽略選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"組合","explanation_zh":"相異性、無放回及不重複均已明示；順序被忽略，所以選擇組合。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306208,"parameters":{"operation":"model_choice","n":4,"k":4,"ordered":false,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":4,\"n\":4,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0009","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 5 個彼此相異物件無放回且不重複選取 0 個，空選取計一種。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306209,"parameters":{"operation":"model_choice","n":5,"k":0,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":0,\"n\":5,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0010","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 5 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。本題忽略選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"組合","explanation_zh":"相異性、無放回及不重複均已明示；順序被忽略，所以選擇組合。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306210,"parameters":{"operation":"model_choice","n":5,"k":1,"ordered":false,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":1,\"n\":5,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0011","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 5 個彼此相異物件無放回且不重複選取 2 個，空選取計一種。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306211,"parameters":{"operation":"model_choice","n":5,"k":2,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":2,\"n\":5,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0012","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 5 個彼此相異物件無放回且不重複選取 5 個，空選取計一種。本題忽略選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"組合","explanation_zh":"相異性、無放回及不重複均已明示；順序被忽略，所以選擇組合。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306212,"parameters":{"operation":"model_choice","n":5,"k":5,"ordered":false,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":5,\"n\":5,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0013","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 6 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306213,"parameters":{"operation":"model_choice","n":6,"k":1,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":1,\"n\":6,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0014","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 6 個彼此相異物件無放回且不重複選取 3 個，空選取計一種。本題忽略選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"組合","explanation_zh":"相異性、無放回及不重複均已明示；順序被忽略，所以選擇組合。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306214,"parameters":{"operation":"model_choice","n":6,"k":3,"ordered":false,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":3,\"n\":6,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0015","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 6 個彼此相異物件無放回且不重複選取 6 個，空選取計一種。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306215,"parameters":{"operation":"model_choice","n":6,"k":6,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":6,\"n\":6,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0016","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 7 個彼此相異物件無放回且不重複選取 2 個，空選取計一種。本題忽略選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"組合","explanation_zh":"相異性、無放回及不重複均已明示；順序被忽略，所以選擇組合。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306216,"parameters":{"operation":"model_choice","n":7,"k":2,"ordered":false,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":2,\"n\":7,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0017","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 7 個彼此相異物件無放回且不重複選取 4 個，空選取計一種。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306217,"parameters":{"operation":"model_choice","n":7,"k":4,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":4,\"n\":7,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0018","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 7 個彼此相異物件無放回且不重複選取 7 個，空選取計一種。本題忽略選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"組合","explanation_zh":"相異性、無放回及不重複均已明示；順序被忽略，所以選擇組合。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306218,"parameters":{"operation":"model_choice","n":7,"k":7,"ordered":false,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":7,\"n\":7,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0019","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 8 個彼此相異物件無放回且不重複選取 0 個，空選取計一種。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306219,"parameters":{"operation":"model_choice","n":8,"k":0,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":0,\"n\":8,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0020","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 8 個彼此相異物件無放回且不重複選取 3 個，空選取計一種。本題忽略選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"組合","explanation_zh":"相異性、無放回及不重複均已明示；順序被忽略，所以選擇組合。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306220,"parameters":{"operation":"model_choice","n":8,"k":3,"ordered":false,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":3,\"n\":8,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0021","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 8 個彼此相異物件無放回且不重複選取 8 個，空選取計一種。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306221,"parameters":{"operation":"model_choice","n":8,"k":8,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":8,\"n\":8,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0022","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 9 個彼此相異物件無放回且不重複選取 1 個，空選取計一種。本題忽略選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"組合","explanation_zh":"相異性、無放回及不重複均已明示；順序被忽略，所以選擇組合。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306222,"parameters":{"operation":"model_choice","n":9,"k":1,"ordered":false,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":1,\"n\":9,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0023","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 9 個彼此相異物件無放回且不重複選取 4 個，空選取計一種。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306223,"parameters":{"operation":"model_choice","n":9,"k":4,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":4,\"n\":9,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0024","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 10 個彼此相異物件無放回且不重複選取 5 個，空選取計一種。本題忽略選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"組合","explanation_zh":"相異性、無放回及不重複均已明示；順序被忽略，所以選擇組合。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306224,"parameters":{"operation":"model_choice","n":10,"k":5,"ordered":false,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":5,\"n\":10,\"operation\":\"model_choice\",\"ordered\":false,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-selection-model-choice-0025","batch_id":"batch-20260930-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"選取模型判定","stem_zh":"從 12 個彼此相異物件無放回且不重複選取 6 個，空選取計一種。本題區分選取順序，應使用排列模型還是組合模型？","answer":{"kind":"short_text","value":"排列","explanation_zh":"相異性、無放回及不重複均已明示；順序有意義，所以選擇排列。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["順序條件","無放回"]},"variant":{"family_id":"qf-selection-model-choice","seed":202609306225,"parameters":{"operation":"model_choice","n":12,"k":6,"ordered":true,"replacement":false,"distinct":true,"semantic_template":"qf-selection-model-choice/{\"distinct\":true,\"k\":6,\"n\":12,\"operation\":\"model_choice\",\"ordered\":true,\"replacement\":false}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0001","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 0 個彼此相異物件無放回且不重複選取 0 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":"1","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 1；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":202609306301,"parameters":{"operation":"quotient","n":0,"k":0,"semantic_template":"qf-permutation-combination-quotient/{\"k\":0,\"n\":0,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0002","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 1 個彼此相異物件無放回且不重複選取 1 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":"1","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 1；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306302,"parameters":{"operation":"quotient","n":1,"k":1,"semantic_template":"qf-permutation-combination-quotient/{\"k\":1,\"n\":1,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0003","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 2 個彼此相異物件無放回且不重複選取 0 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":"1","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 1；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306303,"parameters":{"operation":"quotient","n":2,"k":0,"semantic_template":"qf-permutation-combination-quotient/{\"k\":0,\"n\":2,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0004","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 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。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":202609306304,"parameters":{"operation":"quotient","n":2,"k":2,"semantic_template":"qf-permutation-combination-quotient/{\"k\":2,\"n\":2,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0005","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 3 個彼此相異物件無放回且不重複選取 1 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":"1","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 1；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306305,"parameters":{"operation":"quotient","n":3,"k":1,"semantic_template":"qf-permutation-combination-quotient/{\"k\":1,\"n\":3,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0006","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 3 個彼此相異物件無放回且不重複選取 3 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":"6","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 6；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306306,"parameters":{"operation":"quotient","n":3,"k":3,"semantic_template":"qf-permutation-combination-quotient/{\"k\":3,\"n\":3,\"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"}}
{"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0008","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 4 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":"24","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 24；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306308,"parameters":{"operation":"quotient","n":4,"k":4,"semantic_template":"qf-permutation-combination-quotient/{\"k\":4,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0009","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 5 個彼此相異物件無放回且不重複選取 0 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":"1","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 1；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306309,"parameters":{"operation":"quotient","n":5,"k":0,"semantic_template":"qf-permutation-combination-quotient/{\"k\":0,\"n\":5,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0010","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 5 個彼此相異物件無放回且不重複選取 1 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":"1","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 1；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":202609306310,"parameters":{"operation":"quotient","n":5,"k":1,"semantic_template":"qf-permutation-combination-quotient/{\"k\":1,\"n\":5,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0011","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 5 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":2,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306311,"parameters":{"operation":"quotient","n":5,"k":2,"semantic_template":"qf-permutation-combination-quotient/{\"k\":2,\"n\":5,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0012","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 5 個彼此相異物件無放回且不重複選取 5 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":"120","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 120；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306312,"parameters":{"operation":"quotient","n":5,"k":5,"semantic_template":"qf-permutation-combination-quotient/{\"k\":5,\"n\":5,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0013","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 6 個彼此相異物件無放回且不重複選取 1 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":"1","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 1；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":202609306313,"parameters":{"operation":"quotient","n":6,"k":1,"semantic_template":"qf-permutation-combination-quotient/{\"k\":1,\"n\":6,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0014","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 6 個彼此相異物件無放回且不重複選取 3 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":"6","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 6；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306314,"parameters":{"operation":"quotient","n":6,"k":3,"semantic_template":"qf-permutation-combination-quotient/{\"k\":3,\"n\":6,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0015","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 6 個彼此相異物件無放回且不重複選取 6 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":"720","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 720；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306315,"parameters":{"operation":"quotient","n":6,"k":6,"semantic_template":"qf-permutation-combination-quotient/{\"k\":6,\"n\":6,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0016","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 7 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":202609306316,"parameters":{"operation":"quotient","n":7,"k":2,"semantic_template":"qf-permutation-combination-quotient/{\"k\":2,\"n\":7,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0017","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 7 個彼此相異物件無放回且不重複選取 4 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":"24","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 24；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306317,"parameters":{"operation":"quotient","n":7,"k":4,"semantic_template":"qf-permutation-combination-quotient/{\"k\":4,\"n\":7,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0018","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 7 個彼此相異物件無放回且不重複選取 7 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":"5040","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 5040；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306318,"parameters":{"operation":"quotient","n":7,"k":7,"semantic_template":"qf-permutation-combination-quotient/{\"k\":7,\"n\":7,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0019","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 8 個彼此相異物件無放回且不重複選取 0 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":"1","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 1；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":202609306319,"parameters":{"operation":"quotient","n":8,"k":0,"semantic_template":"qf-permutation-combination-quotient/{\"k\":0,\"n\":8,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0020","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 8 個彼此相異物件無放回且不重複選取 3 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":"6","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 6；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306320,"parameters":{"operation":"quotient","n":8,"k":3,"semantic_template":"qf-permutation-combination-quotient/{\"k\":3,\"n\":8,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0021","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 8 個彼此相異物件無放回且不重複選取 8 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":"40320","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 40320；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306321,"parameters":{"operation":"quotient","n":8,"k":8,"semantic_template":"qf-permutation-combination-quotient/{\"k\":8,\"n\":8,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0022","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 9 個彼此相異物件無放回且不重複選取 1 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":"1","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 1；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":202609306322,"parameters":{"operation":"quotient","n":9,"k":1,"semantic_template":"qf-permutation-combination-quotient/{\"k\":1,\"n\":9,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0023","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 9 個彼此相異物件無放回且不重複選取 4 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":"24","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 24；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306323,"parameters":{"operation":"quotient","n":9,"k":4,"semantic_template":"qf-permutation-combination-quotient/{\"k\":4,\"n\":9,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0024","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 10 個彼此相異物件無放回且不重複選取 5 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":"120","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 120；空集合也恰有一個排序。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["每個子集合的排序","空選取"]},"variant":{"family_id":"qf-permutation-combination-quotient","seed":202609306324,"parameters":{"operation":"quotient","n":10,"k":5,"semantic_template":"qf-permutation-combination-quotient/{\"k\":5,\"n\":10,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260930-r06-permutation-combination-quotient-0025","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 12 個彼此相異物件無放回且不重複選取 6 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":"720","explanation_zh":"每個 k 元子集合有 k! 種排序，因此此商為 720；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":202609306325,"parameters":{"operation":"quotient","n":12,"k":6,"semantic_template":"qf-permutation-combination-quotient/{\"k\":6,\"n\":12,\"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"}}