第 30891 題：含排固定物件的分割

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≥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)。

• G001 — 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)。

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-one-distinguished-counts, density15_selection_one_distinguished_counts, finite_distinct_selection_protocol]

derived = [[total, included_label0, excluded_label0, partition_defect]]

Selection all subsets ordered ORIGINAL available-choice factor0 := mul(1, 16)

Selection all subsets ordered ORIGINAL available-choice factor0 = 16

Selection all subsets ordered ORIGINAL available-choice factor1 := mul(16, 15)

Selection all subsets ordered ORIGINAL available-choice factor1 = 240

Selection all subsets ordered ORIGINAL available-choice factor2 := mul(240, 14)

Selection all subsets ordered ORIGINAL available-choice factor2 = 3360

source roles = [Selection all subsets ordered, 16, 3]

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

Selection all subsets within-subset ORIGINAL factorial factor2 := mul(1, 2)

Selection all subsets within-subset ORIGINAL factorial factor2 = 2

Selection all subsets within-subset ORIGINAL factorial factor3 := mul(2, 3)

Selection all subsets within-subset ORIGINAL factorial factor3 = 6

source roles = [Selection all subsets within-subset, 3]

derived = [6]

Selection all subsets ORIGINAL exact ordered over k factorial := div(3360, 6)

Selection all subsets ORIGINAL exact ordered over k factorial = 560

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

Selection all subsets reverse unordered times k factorial = 3360

source roles = [Selection all subsets, 16, 3, 3360, 6]

derived = [560, 3360]

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 15)

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 = 15

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(15, 14)

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 = 210

source roles = [Selection subsets INCLUDING label0 ordered, 15, 2]

derived = [[15, 14], 210]

Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor2 := mul(1, 2)

Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor2 = 2

source roles = [Selection subsets INCLUDING label0 within-subset, 2]

derived = [2]

Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial := div(210, 2)

Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial = 105

Selection subsets INCLUDING label0 reverse unordered times k factorial := mul(105, 2)

Selection subsets INCLUDING label0 reverse unordered times k factorial = 210

source roles = [Selection subsets INCLUDING label0, 15, 2, 210, 2]

derived = [105, 210]

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 15)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 = 15

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(15, 14)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 = 210

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 := mul(210, 13)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 = 2730

source roles = [Selection subsets EXCLUDING label0 ordered, 15, 3]

derived = [[15, 14, 13], 2730]

Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor2 := mul(1, 2)

Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor2 = 2

Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor3 := mul(2, 3)

Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor3 = 6

source roles = [Selection subsets EXCLUDING label0 within-subset, 3]

derived = [6]

Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial := div(2730, 6)

Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial = 455

Selection subsets EXCLUDING label0 reverse unordered times k factorial := mul(455, 6)

Selection subsets EXCLUDING label0 reverse unordered times k factorial = 2730

source roles = [Selection subsets EXCLUDING label0, 15, 3, 2730, 6]

derived = [455, 2730]

Selection ORIGINAL included plus excluded := sum(105, 455)

Selection ORIGINAL included plus excluded = 560

Selection ORIGINAL partition sum minus whole := sub(560, 560)

Selection ORIGINAL partition sum minus whole = 0

source roles = [16, 3]

derived = [560, 105, 455, 0]

逐步反向核對

• S032 · 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。
• S031 · premise_and_result_audit — 以精確有理數重算 Selection ORIGINAL partition sum minus whole，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Selection ORIGINAL included plus excluded，不以小數近似替代。
• S029 · 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。
• S028 · premise_and_result_audit — 以精確有理數重算 Selection subsets EXCLUDING label0 reverse unordered times k factorial，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial，不以小數近似替代。
• S026 · 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。
• S025 · premise_and_result_audit — 以精確有理數重算 Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor3，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor2，不以小數近似替代。
• S023 · 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。
• S022 · premise_and_result_audit — 以精確有理數重算 Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0，不以小數近似替代。
• S019 · 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。
• S018 · premise_and_result_audit — 以精確有理數重算 Selection subsets INCLUDING label0 reverse unordered times k factorial，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial，不以小數近似替代。
• S016 · 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。
• S015 · premise_and_result_audit — 以精確有理數重算 Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor2，不以小數近似替代。
• S014 · 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。
• S013 · premise_and_result_audit — 以精確有理數重算 Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0，不以小數近似替代。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 Selection all subsets reverse unordered times k factorial，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Selection all subsets ORIGINAL exact ordered over k factorial，不以小數近似替代。
• S008 · 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。
• S007 · premise_and_result_audit — 以精確有理數重算 Selection all subsets within-subset ORIGINAL factorial factor3，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Selection all subsets within-subset ORIGINAL factorial factor2，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 all subsets 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 all subsets 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 all subsets 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

(560, 105, 455, 0)

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。