第 30882 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=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)。

• G001 — 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)。

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, 7)

Selection all subsets ordered ORIGINAL available-choice factor0 = 7

Selection all subsets ordered ORIGINAL available-choice factor1 := mul(7, 6)

Selection all subsets ordered ORIGINAL available-choice factor1 = 42

source roles = [Selection all subsets ordered, 7, 2]

derived = [[7, 6], 42]

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

Selection all subsets within-subset ORIGINAL factorial factor2 = 2

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

derived = [2]

Selection all subsets ORIGINAL exact ordered over k factorial := div(42, 2)

Selection all subsets ORIGINAL exact ordered over k factorial = 21

Selection all subsets reverse unordered times k factorial := mul(21, 2)

Selection all subsets reverse unordered times k factorial = 42

source roles = [Selection all subsets, 7, 2, 42, 2]

derived = [21, 42]

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 6)

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 = 6

source roles = [Selection subsets INCLUDING label0 ordered, 6, 1]

derived = [[6], 6]

source roles = [Selection subsets INCLUDING label0 within-subset, 1]

derived = [1]

Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial := div(6, 1)

Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial = 6

Selection subsets INCLUDING label0 reverse unordered times k factorial := mul(6, 1)

Selection subsets INCLUDING label0 reverse unordered times k factorial = 6

source roles = [Selection subsets INCLUDING label0, 6, 1, 6, 1]

derived = [6, 6]

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 6)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 = 6

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(6, 5)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 = 30

source roles = [Selection subsets EXCLUDING label0 ordered, 6, 2]

derived = [[6, 5], 30]

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

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

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

derived = [2]

Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial := div(30, 2)

Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial = 15

Selection subsets EXCLUDING label0 reverse unordered times k factorial := mul(15, 2)

Selection subsets EXCLUDING label0 reverse unordered times k factorial = 30

source roles = [Selection subsets EXCLUDING label0, 6, 2, 30, 2]

derived = [15, 30]

Selection ORIGINAL included plus excluded := sum(6, 15)

Selection ORIGINAL included plus excluded = 21

Selection ORIGINAL partition sum minus whole := sub(21, 21)

Selection ORIGINAL partition sum minus whole = 0

source roles = [7, 2]

derived = [21, 6, 15, 0]

逐步反向核對

• 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 ORIGINAL partition sum minus whole，不以小數近似替代。
• S024 · 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。
• 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 reverse unordered times k factorial，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 ORIGINAL exact ordered over k factorial，不以小數近似替代。
• S020 · 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。
• S019 · 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。
• 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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1，不以小數近似替代。
• S016 · 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。
• 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 subsets INCLUDING label0 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 subsets INCLUDING label0 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 — 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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor0，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 all subsets reverse unordered times k factorial，不以小數近似替代。
• S007 · 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。
• S006 · 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。
• S005 · 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。
• S004 · 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。
• 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

(21, 6, 15, 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。