{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0001","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=0、k=0，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(1, 1, 1, 是)","explanation_zh":"有序無序排序因子與空選=(1, 1, 1, 是)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156001,"parameters":{"n":0,"k":0,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"1\",\"1\",\"1\",true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0002","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=1、k=1，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(1, 1, 1, 否)","explanation_zh":"有序無序排序因子與空選=(1, 1, 1, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156002,"parameters":{"n":1,"k":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"1\",\"1\",\"1\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0003","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=20、k=19，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(2432902008176640000, 20, 121645100408832000, 否)","explanation_zh":"有序無序排序因子與空選=(2432902008176640000, 20, 121645100408832000, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156003,"parameters":{"n":20,"k":19,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"2432902008176640000\",\"20\",\"121645100408832000\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0004","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=4、k=3，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(24, 4, 6, 否)","explanation_zh":"有序無序排序因子與空選=(24, 4, 6, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156004,"parameters":{"n":4,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"24\",\"4\",\"6\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0005","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=5、k=2，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(20, 10, 2, 否)","explanation_zh":"有序無序排序因子與空選=(20, 10, 2, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156005,"parameters":{"n":5,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"20\",\"10\",\"2\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0006","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=6、k=3，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(120, 20, 6, 否)","explanation_zh":"有序無序排序因子與空選=(120, 20, 6, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156006,"parameters":{"n":6,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"120\",\"20\",\"6\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0007","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=7、k=2，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(42, 21, 2, 否)","explanation_zh":"有序無序排序因子與空選=(42, 21, 2, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156007,"parameters":{"n":7,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"42\",\"21\",\"2\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0008","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=8、k=3，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(336, 56, 6, 否)","explanation_zh":"有序無序排序因子與空選=(336, 56, 6, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156008,"parameters":{"n":8,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"336\",\"56\",\"6\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0009","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=9、k=2，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(72, 36, 2, 否)","explanation_zh":"有序無序排序因子與空選=(72, 36, 2, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156009,"parameters":{"n":9,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"72\",\"36\",\"2\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0010","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=10、k=3，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(720, 120, 6, 否)","explanation_zh":"有序無序排序因子與空選=(720, 120, 6, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156010,"parameters":{"n":10,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"720\",\"120\",\"6\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0011","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=11、k=2，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(110, 55, 2, 否)","explanation_zh":"有序無序排序因子與空選=(110, 55, 2, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156011,"parameters":{"n":11,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"110\",\"55\",\"2\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0012","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=12、k=3，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(1320, 220, 6, 否)","explanation_zh":"有序無序排序因子與空選=(1320, 220, 6, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156012,"parameters":{"n":12,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"1320\",\"220\",\"6\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0013","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=13、k=2，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(156, 78, 2, 否)","explanation_zh":"有序無序排序因子與空選=(156, 78, 2, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156013,"parameters":{"n":13,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"156\",\"78\",\"2\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0014","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=14、k=3，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(2184, 364, 6, 否)","explanation_zh":"有序無序排序因子與空選=(2184, 364, 6, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156014,"parameters":{"n":14,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"2184\",\"364\",\"6\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0015","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=15、k=2，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(210, 105, 2, 否)","explanation_zh":"有序無序排序因子與空選=(210, 105, 2, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156015,"parameters":{"n":15,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"210\",\"105\",\"2\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0016","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=16、k=3，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(3360, 560, 6, 否)","explanation_zh":"有序無序排序因子與空選=(3360, 560, 6, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156016,"parameters":{"n":16,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"3360\",\"560\",\"6\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0017","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=17、k=2，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(272, 136, 2, 否)","explanation_zh":"有序無序排序因子與空選=(272, 136, 2, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156017,"parameters":{"n":17,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"272\",\"136\",\"2\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0018","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=18、k=3，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(4896, 816, 6, 否)","explanation_zh":"有序無序排序因子與空選=(4896, 816, 6, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156018,"parameters":{"n":18,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"4896\",\"816\",\"6\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0019","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=19、k=2，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(342, 171, 2, 否)","explanation_zh":"有序無序排序因子與空選=(342, 171, 2, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156019,"parameters":{"n":19,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"342\",\"171\",\"2\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0020","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=20、k=3，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(6840, 1140, 6, 否)","explanation_zh":"有序無序排序因子與空選=(6840, 1140, 6, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156020,"parameters":{"n":20,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"6840\",\"1140\",\"6\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0021","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=21、k=2，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(420, 210, 2, 否)","explanation_zh":"有序無序排序因子與空選=(420, 210, 2, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156021,"parameters":{"n":21,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"420\",\"210\",\"2\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0022","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=22、k=3，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(9240, 1540, 6, 否)","explanation_zh":"有序無序排序因子與空選=(9240, 1540, 6, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156022,"parameters":{"n":22,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"9240\",\"1540\",\"6\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0023","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=23、k=2，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(506, 253, 2, 否)","explanation_zh":"有序無序排序因子與空選=(506, 253, 2, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156023,"parameters":{"n":23,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"506\",\"253\",\"2\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0024","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=24、k=3，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(12144, 2024, 6, 否)","explanation_zh":"有序無序排序因子與空選=(12144, 2024, 6, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156024,"parameters":{"n":24,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"12144\",\"2024\",\"6\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0025","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"有序無序與排序因子","stem_zh":"相異標記物件無放回，有序與無序分開。n=25、k=2，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。","answer":{"kind":"expression","value":"(600, 300, 2, 否)","explanation_zh":"有序無序排序因子與空選=(600, 300, 2, 否)。以Pascal整數遞推算組合，再乘排序因子得到有序選取數。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有序無序與排序因子"]},"variant":{"family_id":"qf-density15-selection-order-factor-triple","seed":202610156025,"parameters":{"n":25,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_order_factor_triple","semantic_template":"qf-density15-selection-order-factor-triple/[\"600\",\"300\",\"2\",false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0026","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(0)=∅，k=0，model=distinct_without_replacement，候選三列=[[], [0], [0, 0]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 否, 否], [是, 否, 否], [是, 是, 否], [是, 否, 否], [[], 不合選取, 不合選取], [[], 不合選取, 不合選取])","explanation_zh":"三個標記選取候選資格=([是, 否, 否], [是, 否, 否], [是, 是, 否], [是, 否, 否], [[], 不合選取, 不合選取], [[], 不合選取, 不合選取])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156026,"parameters":{"n":0,"k":0,"candidates":[[],[0],[0,0]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[]],[false,false,true,\"invalid\"],[false,false,false,\"invalid\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0027","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(4)=[0, 1, 2, 3]，k=2，model=distinct_without_replacement，候選三列=[[1, 0], [0, 1], [0, 0]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 是], [是, 是, 是], [是, 是, 否], [是, 是, 否], [[1, 0], [0, 1], 不合選取], [[0, 1], [0, 1], 不合選取])","explanation_zh":"三個標記選取候選資格=([是, 是, 是], [是, 是, 是], [是, 是, 否], [是, 是, 否], [[1, 0], [0, 1], 不合選取], [[0, 1], [0, 1], 不合選取])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156027,"parameters":{"n":4,"k":2,"candidates":[[1,0],[0,1],[0,0]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[1,0]],[true,true,true,[0,1]],[true,true,false,\"invalid\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0028","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(4)=[0, 1, 2, 3]，k=2，model=distinct_without_replacement，候選三列=[[0], [0, 1, 2], [0, 4]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([否, 否, 是], [是, 是, 否], [是, 是, 是], [否, 否, 否], [不合選取, 不合選取, 不合選取], [不合選取, 不合選取, 不合選取])","explanation_zh":"三個標記選取候選資格=([否, 否, 是], [是, 是, 否], [是, 是, 是], [否, 否, 否], [不合選取, 不合選取, 不合選取], [不合選取, 不合選取, 不合選取])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156028,"parameters":{"n":4,"k":2,"candidates":[[0],[0,1,2],[0,4]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[false,true,true,\"invalid\"],[true,false,true,\"invalid\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0029","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 1], [0, 1], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 是], [是, 是, 是], [是, 是, 是], [是, 是, 是], [[0, 1], [0, 1], [0, 1]], [[0, 1], [0, 1], [0, 1]])","explanation_zh":"三個標記選取候選資格=([是, 是, 是], [是, 是, 是], [是, 是, 是], [是, 是, 是], [[0, 1], [0, 1], [0, 1]], [[0, 1], [0, 1], [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156029,"parameters":{"n":8,"k":2,"candidates":[[0,1],[0,1],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[0,1]],[true,true,true,[0,1]],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0030","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0], [0, 1], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([否, 是, 是], [是, 是, 是], [是, 是, 是], [否, 是, 是], [不合選取, [0, 1], [0, 1]], [不合選取, [0, 1], [0, 1]])","explanation_zh":"三個標記選取候選資格=([否, 是, 是], [是, 是, 是], [是, 是, 是], [否, 是, 是], [不合選取, [0, 1], [0, 1]], [不合選取, [0, 1], [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156030,"parameters":{"n":8,"k":2,"candidates":[[0],[0,1],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[true,true,true,[0,1]],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0031","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 0], [0, 1], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 是], [是, 是, 是], [否, 是, 是], [否, 是, 是], [不合選取, [0, 1], [0, 1]], [不合選取, [0, 1], [0, 1]])","explanation_zh":"三個標記選取候選資格=([是, 是, 是], [是, 是, 是], [否, 是, 是], [否, 是, 是], [不合選取, [0, 1], [0, 1]], [不合選取, [0, 1], [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156031,"parameters":{"n":8,"k":2,"candidates":[[0,0],[0,1],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,false,\"invalid\"],[true,true,true,[0,1]],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0032","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 8], [0, 1], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 是], [否, 是, 是], [是, 是, 是], [否, 是, 是], [不合選取, [0, 1], [0, 1]], [不合選取, [0, 1], [0, 1]])","explanation_zh":"三個標記選取候選資格=([是, 是, 是], [否, 是, 是], [是, 是, 是], [否, 是, 是], [不合選取, [0, 1], [0, 1]], [不合選取, [0, 1], [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156032,"parameters":{"n":8,"k":2,"candidates":[[0,8],[0,1],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,false,true,\"invalid\"],[true,true,true,[0,1]],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0033","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 1], [0], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 否, 是], [是, 是, 是], [是, 是, 是], [是, 否, 是], [[0, 1], 不合選取, [0, 1]], [[0, 1], 不合選取, [0, 1]])","explanation_zh":"三個標記選取候選資格=([是, 否, 是], [是, 是, 是], [是, 是, 是], [是, 否, 是], [[0, 1], 不合選取, [0, 1]], [[0, 1], 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156033,"parameters":{"n":8,"k":2,"candidates":[[0,1],[0],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[0,1]],[false,true,true,\"invalid\"],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0034","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0], [0], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([否, 否, 是], [是, 是, 是], [是, 是, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])","explanation_zh":"三個標記選取候選資格=([否, 否, 是], [是, 是, 是], [是, 是, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156034,"parameters":{"n":8,"k":2,"candidates":[[0],[0],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[false,true,true,\"invalid\"],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0035","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 0], [0], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 否, 是], [是, 是, 是], [否, 是, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])","explanation_zh":"三個標記選取候選資格=([是, 否, 是], [是, 是, 是], [否, 是, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156035,"parameters":{"n":8,"k":2,"candidates":[[0,0],[0],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,false,\"invalid\"],[false,true,true,\"invalid\"],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0036","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 8], [0], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 否, 是], [否, 是, 是], [是, 是, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])","explanation_zh":"三個標記選取候選資格=([是, 否, 是], [否, 是, 是], [是, 是, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156036,"parameters":{"n":8,"k":2,"candidates":[[0,8],[0],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,false,true,\"invalid\"],[false,true,true,\"invalid\"],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0037","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 1], [0, 0], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 是], [是, 是, 是], [是, 否, 是], [是, 否, 是], [[0, 1], 不合選取, [0, 1]], [[0, 1], 不合選取, [0, 1]])","explanation_zh":"三個標記選取候選資格=([是, 是, 是], [是, 是, 是], [是, 否, 是], [是, 否, 是], [[0, 1], 不合選取, [0, 1]], [[0, 1], 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156037,"parameters":{"n":8,"k":2,"candidates":[[0,1],[0,0],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[0,1]],[true,true,false,\"invalid\"],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0038","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0], [0, 0], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([否, 是, 是], [是, 是, 是], [是, 否, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])","explanation_zh":"三個標記選取候選資格=([否, 是, 是], [是, 是, 是], [是, 否, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156038,"parameters":{"n":8,"k":2,"candidates":[[0],[0,0],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[true,true,false,\"invalid\"],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0039","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 0], [0, 0], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 是], [是, 是, 是], [否, 否, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])","explanation_zh":"三個標記選取候選資格=([是, 是, 是], [是, 是, 是], [否, 否, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156039,"parameters":{"n":8,"k":2,"candidates":[[0,0],[0,0],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,false,\"invalid\"],[true,true,false,\"invalid\"],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0040","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 8], [0, 0], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 是], [否, 是, 是], [是, 否, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])","explanation_zh":"三個標記選取候選資格=([是, 是, 是], [否, 是, 是], [是, 否, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156040,"parameters":{"n":8,"k":2,"candidates":[[0,8],[0,0],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,false,true,\"invalid\"],[true,true,false,\"invalid\"],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0041","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 1], [0, 8], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 是], [是, 否, 是], [是, 是, 是], [是, 否, 是], [[0, 1], 不合選取, [0, 1]], [[0, 1], 不合選取, [0, 1]])","explanation_zh":"三個標記選取候選資格=([是, 是, 是], [是, 否, 是], [是, 是, 是], [是, 否, 是], [[0, 1], 不合選取, [0, 1]], [[0, 1], 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156041,"parameters":{"n":8,"k":2,"candidates":[[0,1],[0,8],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[0,1]],[true,false,true,\"invalid\"],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0042","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0], [0, 8], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([否, 是, 是], [是, 否, 是], [是, 是, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])","explanation_zh":"三個標記選取候選資格=([否, 是, 是], [是, 否, 是], [是, 是, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156042,"parameters":{"n":8,"k":2,"candidates":[[0],[0,8],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[true,false,true,\"invalid\"],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0043","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 0], [0, 8], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 是], [是, 否, 是], [否, 是, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])","explanation_zh":"三個標記選取候選資格=([是, 是, 是], [是, 否, 是], [否, 是, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156043,"parameters":{"n":8,"k":2,"candidates":[[0,0],[0,8],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,false,\"invalid\"],[true,false,true,\"invalid\"],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0044","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 8], [0, 8], [0, 1]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 是], [否, 否, 是], [是, 是, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])","explanation_zh":"三個標記選取候選資格=([是, 是, 是], [否, 否, 是], [是, 是, 是], [否, 否, 是], [不合選取, 不合選取, [0, 1]], [不合選取, 不合選取, [0, 1]])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156044,"parameters":{"n":8,"k":2,"candidates":[[0,8],[0,8],[0,1]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,false,true,\"invalid\"],[true,false,true,\"invalid\"],[true,true,true,[0,1]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0045","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 1], [0, 1], [0]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 否], [是, 是, 是], [是, 是, 是], [是, 是, 否], [[0, 1], [0, 1], 不合選取], [[0, 1], [0, 1], 不合選取])","explanation_zh":"三個標記選取候選資格=([是, 是, 否], [是, 是, 是], [是, 是, 是], [是, 是, 否], [[0, 1], [0, 1], 不合選取], [[0, 1], [0, 1], 不合選取])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156045,"parameters":{"n":8,"k":2,"candidates":[[0,1],[0,1],[0]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[0,1]],[true,true,true,[0,1]],[false,true,true,\"invalid\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0046","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0], [0, 1], [0]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([否, 是, 否], [是, 是, 是], [是, 是, 是], [否, 是, 否], [不合選取, [0, 1], 不合選取], [不合選取, [0, 1], 不合選取])","explanation_zh":"三個標記選取候選資格=([否, 是, 否], [是, 是, 是], [是, 是, 是], [否, 是, 否], [不合選取, [0, 1], 不合選取], [不合選取, [0, 1], 不合選取])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156046,"parameters":{"n":8,"k":2,"candidates":[[0],[0,1],[0]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[true,true,true,[0,1]],[false,true,true,\"invalid\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0047","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 0], [0, 1], [0]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 否], [是, 是, 是], [否, 是, 是], [否, 是, 否], [不合選取, [0, 1], 不合選取], [不合選取, [0, 1], 不合選取])","explanation_zh":"三個標記選取候選資格=([是, 是, 否], [是, 是, 是], [否, 是, 是], [否, 是, 否], [不合選取, [0, 1], 不合選取], [不合選取, [0, 1], 不合選取])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156047,"parameters":{"n":8,"k":2,"candidates":[[0,0],[0,1],[0]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,false,\"invalid\"],[true,true,true,[0,1]],[false,true,true,\"invalid\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0048","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 8], [0, 1], [0]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 是, 否], [否, 是, 是], [是, 是, 是], [否, 是, 否], [不合選取, [0, 1], 不合選取], [不合選取, [0, 1], 不合選取])","explanation_zh":"三個標記選取候選資格=([是, 是, 否], [否, 是, 是], [是, 是, 是], [否, 是, 否], [不合選取, [0, 1], 不合選取], [不合選取, [0, 1], 不合選取])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156048,"parameters":{"n":8,"k":2,"candidates":[[0,8],[0,1],[0]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,false,true,\"invalid\"],[true,true,true,[0,1]],[false,true,true,\"invalid\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0049","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0, 1], [0], [0]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([是, 否, 否], [是, 是, 是], [是, 是, 是], [是, 否, 否], [[0, 1], 不合選取, 不合選取], [[0, 1], 不合選取, 不合選取])","explanation_zh":"三個標記選取候選資格=([是, 否, 否], [是, 是, 是], [是, 是, 是], [是, 否, 否], [[0, 1], 不合選取, 不合選取], [[0, 1], 不合選取, 不合選取])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156049,"parameters":{"n":8,"k":2,"candidates":[[0,1],[0],[0]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[true,true,true,[0,1]],[false,true,true,\"invalid\"],[false,true,true,\"invalid\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0050","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"三個選取候選的資格軸","stem_zh":"相異標記物件無放回，有序與無序分開。Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，k=2，model=distinct_without_replacement，候選三列=[[0], [0], [0]]。依原序求 (長度資格, 全標記域內資格, 全相異資格, 完整選取資格, 合格有序tuple或不合選取, 合格排序subset或不合選取)。","answer":{"kind":"expression","value":"([否, 否, 否], [是, 是, 是], [是, 是, 是], [否, 否, 否], [不合選取, 不合選取, 不合選取], [不合選取, 不合選取, 不合選取])","explanation_zh":"三個標記選取候選資格=([否, 否, 否], [是, 是, 是], [是, 是, 是], [否, 否, 否], [不合選取, 不合選取, 不合選取], [不合選取, 不合選取, 不合選取])。長度、型別、相異三軸分開核；合格列才保留有序列和其無序版本。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個選取候選的資格軸"]},"variant":{"family_id":"qf-density15-selection-fixed-label-qualification","seed":202610156050,"parameters":{"n":8,"k":2,"candidates":[[0],[0],[0]],"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_fixed_label_qualification","semantic_template":"qf-density15-selection-fixed-label-qualification/[[false,true,true,\"invalid\"],[false,true,true,\"invalid\"],[false,true,true,\"invalid\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0051","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=1≥1、k=0、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(1, 0, 1, 0)","explanation_zh":"含排指定物件組合分割=(1, 0, 1, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156051,"parameters":{"n":1,"k":0,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"1\",\"0\",\"1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0052","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=3≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(1, 1, 0, 0)","explanation_zh":"含排指定物件組合分割=(1, 1, 0, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156052,"parameters":{"n":3,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"1\",\"1\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0053","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=20≥1、k=10、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(184756, 92378, 92378, 0)","explanation_zh":"含排指定物件組合分割=(184756, 92378, 92378, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156053,"parameters":{"n":20,"k":10,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"184756\",\"92378\",\"92378\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0054","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=6≥1、k=5、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(6, 5, 1, 0)","explanation_zh":"含排指定物件組合分割=(6, 5, 1, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156054,"parameters":{"n":6,"k":5,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"6\",\"5\",\"1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0055","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=5≥1、k=2、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(10, 4, 6, 0)","explanation_zh":"含排指定物件組合分割=(10, 4, 6, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156055,"parameters":{"n":5,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"10\",\"4\",\"6\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0056","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=6≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(20, 10, 10, 0)","explanation_zh":"含排指定物件組合分割=(20, 10, 10, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156056,"parameters":{"n":6,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"20\",\"10\",\"10\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0057","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=7≥1、k=2、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(21, 6, 15, 0)","explanation_zh":"含排指定物件組合分割=(21, 6, 15, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156057,"parameters":{"n":7,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"21\",\"6\",\"15\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0058","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=8≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(56, 21, 35, 0)","explanation_zh":"含排指定物件組合分割=(56, 21, 35, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156058,"parameters":{"n":8,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"56\",\"21\",\"35\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0059","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=9≥1、k=2、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(36, 8, 28, 0)","explanation_zh":"含排指定物件組合分割=(36, 8, 28, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156059,"parameters":{"n":9,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"36\",\"8\",\"28\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0060","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=10≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(120, 36, 84, 0)","explanation_zh":"含排指定物件組合分割=(120, 36, 84, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156060,"parameters":{"n":10,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"120\",\"36\",\"84\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0061","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=11≥1、k=2、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(55, 10, 45, 0)","explanation_zh":"含排指定物件組合分割=(55, 10, 45, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156061,"parameters":{"n":11,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"55\",\"10\",\"45\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0062","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=12≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(220, 55, 165, 0)","explanation_zh":"含排指定物件組合分割=(220, 55, 165, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156062,"parameters":{"n":12,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"220\",\"55\",\"165\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0063","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=13≥1、k=2、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(78, 12, 66, 0)","explanation_zh":"含排指定物件組合分割=(78, 12, 66, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156063,"parameters":{"n":13,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"78\",\"12\",\"66\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0064","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=14≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(364, 78, 286, 0)","explanation_zh":"含排指定物件組合分割=(364, 78, 286, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156064,"parameters":{"n":14,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"364\",\"78\",\"286\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0065","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=15≥1、k=2、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(105, 14, 91, 0)","explanation_zh":"含排指定物件組合分割=(105, 14, 91, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156065,"parameters":{"n":15,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"105\",\"14\",\"91\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0066","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=16≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(560, 105, 455, 0)","explanation_zh":"含排指定物件組合分割=(560, 105, 455, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156066,"parameters":{"n":16,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"560\",\"105\",\"455\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0067","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=17≥1、k=2、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(136, 16, 120, 0)","explanation_zh":"含排指定物件組合分割=(136, 16, 120, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156067,"parameters":{"n":17,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"136\",\"16\",\"120\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0068","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=18≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(816, 136, 680, 0)","explanation_zh":"含排指定物件組合分割=(816, 136, 680, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156068,"parameters":{"n":18,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"816\",\"136\",\"680\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0069","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=19≥1、k=2、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(171, 18, 153, 0)","explanation_zh":"含排指定物件組合分割=(171, 18, 153, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156069,"parameters":{"n":19,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"171\",\"18\",\"153\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0070","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=20≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(1140, 171, 969, 0)","explanation_zh":"含排指定物件組合分割=(1140, 171, 969, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156070,"parameters":{"n":20,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"1140\",\"171\",\"969\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0071","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=21≥1、k=2、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(210, 20, 190, 0)","explanation_zh":"含排指定物件組合分割=(210, 20, 190, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156071,"parameters":{"n":21,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"210\",\"20\",\"190\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0072","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=22≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(1540, 210, 1330, 0)","explanation_zh":"含排指定物件組合分割=(1540, 210, 1330, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156072,"parameters":{"n":22,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"1540\",\"210\",\"1330\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0073","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=23≥1、k=2、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(253, 22, 231, 0)","explanation_zh":"含排指定物件組合分割=(253, 22, 231, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156073,"parameters":{"n":23,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"253\",\"22\",\"231\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0074","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=24≥1、k=3、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(2024, 253, 1771, 0)","explanation_zh":"含排指定物件組合分割=(2024, 253, 1771, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156074,"parameters":{"n":24,"k":3,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"2024\",\"253\",\"1771\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0075","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"含排固定物件的分割","stem_zh":"相異標記物件無放回，有序與無序分開。n=25≥1、k=2、model=distinct_without_replacement，指定已存在標記0。求 (全組合數, 含0組合數, 排0組合數, 兩部分和減全數)。","answer":{"kind":"expression","value":"(300, 24, 276, 0)","explanation_zh":"含排指定物件組合分割=(300, 24, 276, 0)。移出固定物件後以Pascal列分別核兩個合法選取大小；k=0含0數為零。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["含排固定物件的分割"]},"variant":{"family_id":"qf-density15-selection-one-distinguished-counts","seed":202610156075,"parameters":{"n":25,"k":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_one_distinguished_counts","semantic_template":"qf-density15-selection-one-distinguished-counts/[\"300\",\"24\",\"276\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0076","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=0、k=0、r=0、model=distinct_without_replacement，前r個位置固定原序標記=[]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(0, 0, 1, 1, 1/1)","explanation_zh":"固定前序餘量與完成計數比=(0, 0, 1, 1, 1/1)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156076,"parameters":{"n":0,"k":0,"r":0,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[0,0,\"1\",\"1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0077","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=4、k=2、r=0、model=distinct_without_replacement，前r個位置固定原序標記=[]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(4, 2, 12, 12, 1/1)","explanation_zh":"固定前序餘量與完成計數比=(4, 2, 12, 12, 1/1)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156077,"parameters":{"n":4,"k":2,"r":0,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[4,2,\"12\",\"12\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0078","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=4、k=2、r=2、model=distinct_without_replacement，前r個位置固定原序標記=[0, 1]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(2, 0, 1, 12, 1/12)","explanation_zh":"固定前序餘量與完成計數比=(2, 0, 1, 12, 1/12)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156078,"parameters":{"n":4,"k":2,"r":2,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[2,0,\"1\",\"12\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0079","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=32、k=32、r=16、model=distinct_without_replacement，前r個位置固定原序標記=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(16, 16, 20922789888000, 263130836933693530167218012160000000, 1/12576278705767096320000)","explanation_zh":"固定前序餘量與完成計數比=(16, 16, 20922789888000, 263130836933693530167218012160000000, 1/12576278705767096320000)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156079,"parameters":{"n":32,"k":32,"r":16,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[16,16,\"20922789888000\",\"263130836933693530167218012160000000\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0080","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=5、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(4, 2, 12, 60, 1/5)","explanation_zh":"固定前序餘量與完成計數比=(4, 2, 12, 60, 1/5)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156080,"parameters":{"n":5,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[4,2,\"12\",\"60\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0081","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=6、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(5, 2, 20, 120, 1/6)","explanation_zh":"固定前序餘量與完成計數比=(5, 2, 20, 120, 1/6)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156081,"parameters":{"n":6,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[5,2,\"20\",\"120\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0082","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=7、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(6, 2, 30, 210, 1/7)","explanation_zh":"固定前序餘量與完成計數比=(6, 2, 30, 210, 1/7)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156082,"parameters":{"n":7,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[6,2,\"30\",\"210\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0083","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=8、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(7, 2, 42, 336, 1/8)","explanation_zh":"固定前序餘量與完成計數比=(7, 2, 42, 336, 1/8)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156083,"parameters":{"n":8,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[7,2,\"42\",\"336\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0084","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=9、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(8, 2, 56, 504, 1/9)","explanation_zh":"固定前序餘量與完成計數比=(8, 2, 56, 504, 1/9)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156084,"parameters":{"n":9,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[8,2,\"56\",\"504\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0085","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=10、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(9, 2, 72, 720, 1/10)","explanation_zh":"固定前序餘量與完成計數比=(9, 2, 72, 720, 1/10)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156085,"parameters":{"n":10,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[9,2,\"72\",\"720\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0086","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=11、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(10, 2, 90, 990, 1/11)","explanation_zh":"固定前序餘量與完成計數比=(10, 2, 90, 990, 1/11)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156086,"parameters":{"n":11,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[10,2,\"90\",\"990\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0087","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=12、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(11, 2, 110, 1320, 1/12)","explanation_zh":"固定前序餘量與完成計數比=(11, 2, 110, 1320, 1/12)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156087,"parameters":{"n":12,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[11,2,\"110\",\"1320\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0088","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=13、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(12, 2, 132, 1716, 1/13)","explanation_zh":"固定前序餘量與完成計數比=(12, 2, 132, 1716, 1/13)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156088,"parameters":{"n":13,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[12,2,\"132\",\"1716\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0089","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=14、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(13, 2, 156, 2184, 1/14)","explanation_zh":"固定前序餘量與完成計數比=(13, 2, 156, 2184, 1/14)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156089,"parameters":{"n":14,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[13,2,\"156\",\"2184\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0090","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=15、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(14, 2, 182, 2730, 1/15)","explanation_zh":"固定前序餘量與完成計數比=(14, 2, 182, 2730, 1/15)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156090,"parameters":{"n":15,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[14,2,\"182\",\"2730\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0091","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=16、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(15, 2, 210, 3360, 1/16)","explanation_zh":"固定前序餘量與完成計數比=(15, 2, 210, 3360, 1/16)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156091,"parameters":{"n":16,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[15,2,\"210\",\"3360\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0092","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=17、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(16, 2, 240, 4080, 1/17)","explanation_zh":"固定前序餘量與完成計數比=(16, 2, 240, 4080, 1/17)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156092,"parameters":{"n":17,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[16,2,\"240\",\"4080\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0093","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=18、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(17, 2, 272, 4896, 1/18)","explanation_zh":"固定前序餘量與完成計數比=(17, 2, 272, 4896, 1/18)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156093,"parameters":{"n":18,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[17,2,\"272\",\"4896\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0094","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=19、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(18, 2, 306, 5814, 1/19)","explanation_zh":"固定前序餘量與完成計數比=(18, 2, 306, 5814, 1/19)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156094,"parameters":{"n":19,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[18,2,\"306\",\"5814\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0095","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=20、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(19, 2, 342, 6840, 1/20)","explanation_zh":"固定前序餘量與完成計數比=(19, 2, 342, 6840, 1/20)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156095,"parameters":{"n":20,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[19,2,\"342\",\"6840\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0096","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=21、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(20, 2, 380, 7980, 1/21)","explanation_zh":"固定前序餘量與完成計數比=(20, 2, 380, 7980, 1/21)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156096,"parameters":{"n":21,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[20,2,\"380\",\"7980\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0097","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=22、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(21, 2, 420, 9240, 1/22)","explanation_zh":"固定前序餘量與完成計數比=(21, 2, 420, 9240, 1/22)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156097,"parameters":{"n":22,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[21,2,\"420\",\"9240\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0098","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=23、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(22, 2, 462, 10626, 1/23)","explanation_zh":"固定前序餘量與完成計數比=(22, 2, 462, 10626, 1/23)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156098,"parameters":{"n":23,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[22,2,\"462\",\"10626\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0099","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=24、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(23, 2, 506, 12144, 1/24)","explanation_zh":"固定前序餘量與完成計數比=(23, 2, 506, 12144, 1/24)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156099,"parameters":{"n":24,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[23,2,\"506\",\"12144\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r06-permutation-combination-0100","batch_id":"batch-20261011-advance-20261015-r06-permutation-combination-foundations","language":"zh-Hant","source_mko_ids":["mko-permutation-combination"],"title_zh":"固定前序的完成計數","stem_zh":"相異標記物件無放回，有序與無序分開。n=25、k=3、r=1、model=distinct_without_replacement，前r個位置固定原序標記=[0]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。","answer":{"kind":"expression","value":"(24, 2, 552, 13800, 1/25)","explanation_zh":"固定前序餘量與完成計數比=(24, 2, 552, 13800, 1/25)。固定前序移去已用物件與位置，再核兩個有序計數並規約其比。比例只作計數比不自行稱概率。空選取只有一種，不套重複不可區分或環形模型，有限計數非一般形式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["固定前序的完成計數"]},"variant":{"family_id":"qf-density15-selection-forced-initial-order","seed":202610156100,"parameters":{"n":25,"k":3,"r":1,"model":"distinct_without_replacement","domain":"finite_distinct_selection_protocol","operation":"density15_selection_forced_initial_order","semantic_template":"qf-density15-selection-forced-initial-order/[24,2,\"552\",\"13800\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}