第 30887 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=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)。

• G001 — 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)。

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, 12)

Selection all subsets ordered ORIGINAL available-choice factor0 = 12

Selection all subsets ordered ORIGINAL available-choice factor1 := mul(12, 11)

Selection all subsets ordered ORIGINAL available-choice factor1 = 132

Selection all subsets ordered ORIGINAL available-choice factor2 := mul(132, 10)

Selection all subsets ordered ORIGINAL available-choice factor2 = 1320

source roles = [Selection all subsets ordered, 12, 3]

derived = [[12, 11, 10], 1320]

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(1320, 6)

Selection all subsets ORIGINAL exact ordered over k factorial = 220

Selection all subsets reverse unordered times k factorial := mul(220, 6)

Selection all subsets reverse unordered times k factorial = 1320

source roles = [Selection all subsets, 12, 3, 1320, 6]

derived = [220, 1320]

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 11)

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 = 11

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(11, 10)

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 = 110

source roles = [Selection subsets INCLUDING label0 ordered, 11, 2]

derived = [[11, 10], 110]

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(110, 2)

Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial = 55

Selection subsets INCLUDING label0 reverse unordered times k factorial := mul(55, 2)

Selection subsets INCLUDING label0 reverse unordered times k factorial = 110

source roles = [Selection subsets INCLUDING label0, 11, 2, 110, 2]

derived = [55, 110]

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 11)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 = 11

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(11, 10)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 = 110

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 := mul(110, 9)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 = 990

source roles = [Selection subsets EXCLUDING label0 ordered, 11, 3]

derived = [[11, 10, 9], 990]

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(990, 6)

Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial = 165

Selection subsets EXCLUDING label0 reverse unordered times k factorial := mul(165, 6)

Selection subsets EXCLUDING label0 reverse unordered times k factorial = 990

source roles = [Selection subsets EXCLUDING label0, 11, 3, 990, 6]

derived = [165, 990]

Selection ORIGINAL included plus excluded := sum(55, 165)

Selection ORIGINAL included plus excluded = 220

Selection ORIGINAL partition sum minus whole := sub(220, 220)

Selection ORIGINAL partition sum minus whole = 0

source roles = [12, 3]

derived = [220, 55, 165, 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

(220, 55, 165, 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。