第 30916 題：有序無序與排序因子

原題與已知條件

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)。

• G001 — 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)。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-permutation-combination

逐步推導

source roles = [qf-density15-selection-order-factor-triple, density15_selection_order_factor_triple, finite_distinct_selection_protocol]

derived = [[ordered_count, unordered_count, ordering_factor, empty_selection]]

Selection original full ordered ORIGINAL available-choice factor0 := mul(1, 16)

Selection original full ordered ORIGINAL available-choice factor0 = 16

Selection original full ordered ORIGINAL available-choice factor1 := mul(16, 15)

Selection original full ordered ORIGINAL available-choice factor1 = 240

Selection original full ordered ORIGINAL available-choice factor2 := mul(240, 14)

Selection original full ordered ORIGINAL available-choice factor2 = 3360

source roles = [Selection original full ordered, 16, 3]

derived = [[16, 15, 14], 3360]

Selection original unordered ordered ORIGINAL available-choice factor0 := mul(1, 16)

Selection original unordered ordered ORIGINAL available-choice factor0 = 16

Selection original unordered ordered ORIGINAL available-choice factor1 := mul(16, 15)

Selection original unordered ordered ORIGINAL available-choice factor1 = 240

Selection original unordered ordered ORIGINAL available-choice factor2 := mul(240, 14)

Selection original unordered ordered ORIGINAL available-choice factor2 = 3360

source roles = [Selection original unordered ordered, 16, 3]

derived = [[16, 15, 14], 3360]

Selection original unordered within-subset ORIGINAL factorial factor2 := mul(1, 2)

Selection original unordered within-subset ORIGINAL factorial factor2 = 2

Selection original unordered within-subset ORIGINAL factorial factor3 := mul(2, 3)

Selection original unordered within-subset ORIGINAL factorial factor3 = 6

source roles = [Selection original unordered within-subset, 3]

derived = [6]

Selection original unordered ORIGINAL exact ordered over k factorial := div(3360, 6)

Selection original unordered ORIGINAL exact ordered over k factorial = 560

Selection original unordered reverse unordered times k factorial := mul(560, 6)

Selection original unordered reverse unordered times k factorial = 3360

source roles = [Selection original unordered, 16, 3, 3360, 6]

derived = [560, 3360]

Selection original ordering factor ORIGINAL factorial factor2 := mul(1, 2)

Selection original ordering factor ORIGINAL factorial factor2 = 2

Selection original ordering factor ORIGINAL factorial factor3 := mul(2, 3)

Selection original ordering factor ORIGINAL factorial factor3 = 6

source roles = [Selection original ordering factor, 3]

derived = [6]

逐步反向核對

• S018 · premise_and_result_audit — 用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。
• S017 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor3，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor2，不以小數近似替代。
• S015 · premise_and_result_audit — 用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。
• S014 · premise_and_result_audit — 以精確有理數重算 Selection original unordered reverse unordered times k factorial，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ORIGINAL exact ordered over k factorial，不以小數近似替代。
• S012 · premise_and_result_audit — 用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。
• S011 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor3，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor2，不以小數近似替代。
• S009 · premise_and_result_audit — 用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。
• S008 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor2，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor1，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor0，不以小數近似替代。
• S005 · premise_and_result_audit — 用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。
• S004 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor2，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor1，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor0，不以小數近似替代。
• S001 · premise_and_result_audit — 用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。

結論

(3360, 560, 6, 否)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。