第 6770 題：省去一次因子的乘積

原題與已知條件

N從0起；BigInt精確階乘；n=21、k=20∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。

• G001 — N從0起；BigInt精確階乘；n=21、k=20∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。

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-factorial

逐步推導

N includes0;F(0)=1;F(j)=j·F(j−1)

exact integer products,not floating values or a permutation proof

actual factorial:start at F(0)

F(0)=1

actual factorial through factor1 := mul(1, 1)

actual factorial through factor1 = 1

actual factorial through factor2 := mul(1, 2)

actual factorial through factor2 = 2

actual factorial through factor3 := mul(2, 3)

actual factorial through factor3 = 6

actual factorial through factor4 := mul(6, 4)

actual factorial through factor4 = 24

actual factorial through factor5 := mul(24, 5)

actual factorial through factor5 = 120

actual factorial through factor6 := mul(120, 6)

actual factorial through factor6 = 720

actual factorial through factor7 := mul(720, 7)

actual factorial through factor7 = 5040

actual factorial through factor8 := mul(5040, 8)

actual factorial through factor8 = 40320

actual factorial through factor9 := mul(40320, 9)

actual factorial through factor9 = 362880

actual factorial through factor10 := mul(362880, 10)

actual factorial through factor10 = 3628800

actual factorial through factor11 := mul(3628800, 11)

actual factorial through factor11 = 39916800

actual factorial through factor12 := mul(39916800, 12)

actual factorial through factor12 = 479001600

actual factorial through factor13 := mul(479001600, 13)

actual factorial through factor13 = 6227020800

actual factorial through factor14 := mul(6227020800, 14)

actual factorial through factor14 = 87178291200

actual factorial through factor15 := mul(87178291200, 15)

actual factorial through factor15 = 1307674368000

actual factorial through factor16 := mul(1307674368000, 16)

actual factorial through factor16 = 20922789888000

actual factorial through factor17 := mul(20922789888000, 17)

actual factorial through factor17 = 355687428096000

actual factorial through factor18 := mul(355687428096000, 18)

actual factorial through factor18 = 6402373705728000

actual factorial through factor19 := mul(6402373705728000, 19)

actual factorial through factor19 = 121645100408832000

actual factorial through factor20 := mul(121645100408832000, 20)

actual factorial through factor20 = 2432902008176640000

actual factorial through factor21 := mul(2432902008176640000, 21)

actual factorial through factor21 = 51090942171709440000

single-factor-omitted product := div(51090942171709440000, 20)

single-factor-omitted product = 2554547108585472000

restore omitted factor := mul(20, 2554547108585472000)

restore omitted factor = 51090942171709440000

restored product minus factorial := sub(51090942171709440000, 51090942171709440000)

restored product minus factorial = 0

removed factor k=20;full n!=51090942171709440000

omitted product=2554547108585472000;restoration residual=0

逐步反向核對

• S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S026 · premise_and_result_audit — 以精確有理數重算 restored product minus factorial，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 restore omitted factor，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 single-factor-omitted product，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor21，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor20，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor19，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor18，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor17，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor16，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor15，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor14，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor13，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor12，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor11，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor10，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor9，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor8，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor7，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor6，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor5，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor4，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor3，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor2，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 actual factorial through factor1，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(51090942171709440000, 2554547108585472000, 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。