第 30879 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=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)。

• G001 — 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)。

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, 6)

Selection all subsets ordered ORIGINAL available-choice factor0 = 6

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

Selection all subsets ordered ORIGINAL available-choice factor1 = 30

Selection all subsets ordered ORIGINAL available-choice factor2 := mul(30, 4)

Selection all subsets ordered ORIGINAL available-choice factor2 = 120

Selection all subsets ordered ORIGINAL available-choice factor3 := mul(120, 3)

Selection all subsets ordered ORIGINAL available-choice factor3 = 360

Selection all subsets ordered ORIGINAL available-choice factor4 := mul(360, 2)

Selection all subsets ordered ORIGINAL available-choice factor4 = 720

source roles = [Selection all subsets ordered, 6, 5]

derived = [[6, 5, 4, 3, 2], 720]

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

Selection all subsets within-subset ORIGINAL factorial factor4 := mul(6, 4)

Selection all subsets within-subset ORIGINAL factorial factor4 = 24

Selection all subsets within-subset ORIGINAL factorial factor5 := mul(24, 5)

Selection all subsets within-subset ORIGINAL factorial factor5 = 120

source roles = [Selection all subsets within-subset, 5]

derived = [120]

Selection all subsets ORIGINAL exact ordered over k factorial := div(720, 120)

Selection all subsets ORIGINAL exact ordered over k factorial = 6

Selection all subsets reverse unordered times k factorial := mul(6, 120)

Selection all subsets reverse unordered times k factorial = 720

source roles = [Selection all subsets, 6, 5, 720, 120]

derived = [6, 720]

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 5)

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 = 5

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(5, 4)

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 = 20

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor2 := mul(20, 3)

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor2 = 60

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor3 := mul(60, 2)

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor3 = 120

source roles = [Selection subsets INCLUDING label0 ordered, 5, 4]

derived = [[5, 4, 3, 2], 120]

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

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

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

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

Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor4 := mul(6, 4)

Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor4 = 24

source roles = [Selection subsets INCLUDING label0 within-subset, 4]

derived = [24]

Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial := div(120, 24)

Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial = 5

Selection subsets INCLUDING label0 reverse unordered times k factorial := mul(5, 24)

Selection subsets INCLUDING label0 reverse unordered times k factorial = 120

source roles = [Selection subsets INCLUDING label0, 5, 4, 120, 24]

derived = [5, 120]

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 5)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 = 5

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(5, 4)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 = 20

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 := mul(20, 3)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 = 60

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor3 := mul(60, 2)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor3 = 120

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor4 := mul(120, 1)

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor4 = 120

source roles = [Selection subsets EXCLUDING label0 ordered, 5, 5]

derived = [[5, 4, 3, 2, 1], 120]

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

Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor4 := mul(6, 4)

Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor4 = 24

Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor5 := mul(24, 5)

Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor5 = 120

source roles = [Selection subsets EXCLUDING label0 within-subset, 5]

derived = [120]

Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial := div(120, 120)

Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial = 1

Selection subsets EXCLUDING label0 reverse unordered times k factorial := mul(1, 120)

Selection subsets EXCLUDING label0 reverse unordered times k factorial = 120

source roles = [Selection subsets EXCLUDING label0, 5, 5, 120, 120]

derived = [1, 120]

Selection ORIGINAL included plus excluded := sum(5, 1)

Selection ORIGINAL included plus excluded = 6

Selection ORIGINAL partition sum minus whole := sub(6, 6)

Selection ORIGINAL partition sum minus whole = 0

source roles = [6, 5]

derived = [6, 5, 1, 0]

逐步反向核對

• S044 · 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。
• S043 · 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。
• S042 · 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。
• S041 · 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。
• S040 · 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。
• S039 · 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。
• S038 · 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。
• S037 · premise_and_result_audit — 以精確有理數重算 Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor5，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor4，不以小數近似替代。
• S035 · 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。
• S034 · 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。
• S033 · 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。
• S032 · premise_and_result_audit — 以精確有理數重算 Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor4，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor3，不以小數近似替代。
• S030 · 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。
• S029 · 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。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• S025 · 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。
• S024 · 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。
• S023 · premise_and_result_audit — 以精確有理數重算 Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor4，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor3，不以小數近似替代。
• S021 · 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。
• 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 INCLUDING label0 ordered ORIGINAL available-choice factor3，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor2，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 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 INCLUDING 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 all subsets 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 all subsets 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 all subsets within-subset ORIGINAL factorial factor5，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Selection all subsets within-subset ORIGINAL factorial factor4，不以小數近似替代。
• S009 · 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。
• S008 · 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。
• S007 · 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。
• S006 · premise_and_result_audit — 以精確有理數重算 Selection all subsets ordered ORIGINAL available-choice factor4，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Selection all subsets ordered ORIGINAL available-choice factor3，不以小數近似替代。
• 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

(6, 5, 1, 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。