R0060-30903 · q-20261011-advance-20261015-r06-permutation-combination-0003
相異標記物件無放回，有序與無序分開。n=20、k=19，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。
G001 — 相異標記物件無放回，有序與無序分開。n=20、k=19，model=distinct_without_replacement。求 (P(n,k), C(n,k), k!, k=0空選資格)。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。
操作前
source roles = [qf-density15-selection-order-factor-triple, density15_selection_order_factor_triple, finite_distinct_selection_protocol]
操作後
derived = [[ordered_count, unordered_count, ordering_factor, empty_selection]]
條件：無
density14/15用full-prefix underscores，density17用local hyphens。coordinate frame與circle frame不同，輸出欄序／型別依本family，不改原中英／JSON／status。
規則：R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · 依賴：G001
操作前
Selection original full ordered ORIGINAL available-choice factor0 := mul(1, 20)
操作後
Selection original full ordered ORIGINAL available-choice factor0 = 20
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S001
操作前
Selection original full ordered ORIGINAL available-choice factor1 := mul(20, 19)
操作後
Selection original full ordered ORIGINAL available-choice factor1 = 380
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S002
操作前
Selection original full ordered ORIGINAL available-choice factor2 := mul(380, 18)
操作後
Selection original full ordered ORIGINAL available-choice factor2 = 6840
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S003
操作前
Selection original full ordered ORIGINAL available-choice factor3 := mul(6840, 17)
操作後
Selection original full ordered ORIGINAL available-choice factor3 = 116280
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S004
操作前
Selection original full ordered ORIGINAL available-choice factor4 := mul(116280, 16)
操作後
Selection original full ordered ORIGINAL available-choice factor4 = 1860480
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S005
操作前
Selection original full ordered ORIGINAL available-choice factor5 := mul(1860480, 15)
操作後
Selection original full ordered ORIGINAL available-choice factor5 = 27907200
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S006
操作前
Selection original full ordered ORIGINAL available-choice factor6 := mul(27907200, 14)
操作後
Selection original full ordered ORIGINAL available-choice factor6 = 390700800
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S007
操作前
Selection original full ordered ORIGINAL available-choice factor7 := mul(390700800, 13)
操作後
Selection original full ordered ORIGINAL available-choice factor7 = 5079110400
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S008
操作前
Selection original full ordered ORIGINAL available-choice factor8 := mul(5079110400, 12)
操作後
Selection original full ordered ORIGINAL available-choice factor8 = 60949324800
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S009
操作前
Selection original full ordered ORIGINAL available-choice factor9 := mul(60949324800, 11)
操作後
Selection original full ordered ORIGINAL available-choice factor9 = 670442572800
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S010
操作前
Selection original full ordered ORIGINAL available-choice factor10 := mul(670442572800, 10)
操作後
Selection original full ordered ORIGINAL available-choice factor10 = 6704425728000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S011
操作前
Selection original full ordered ORIGINAL available-choice factor11 := mul(6704425728000, 9)
操作後
Selection original full ordered ORIGINAL available-choice factor11 = 60339831552000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S012
操作前
Selection original full ordered ORIGINAL available-choice factor12 := mul(60339831552000, 8)
操作後
Selection original full ordered ORIGINAL available-choice factor12 = 482718652416000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S013
操作前
Selection original full ordered ORIGINAL available-choice factor13 := mul(482718652416000, 7)
操作後
Selection original full ordered ORIGINAL available-choice factor13 = 3379030566912000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S014
操作前
Selection original full ordered ORIGINAL available-choice factor14 := mul(3379030566912000, 6)
操作後
Selection original full ordered ORIGINAL available-choice factor14 = 20274183401472000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S015
操作前
Selection original full ordered ORIGINAL available-choice factor15 := mul(20274183401472000, 5)
操作後
Selection original full ordered ORIGINAL available-choice factor15 = 101370917007360000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S016
操作前
Selection original full ordered ORIGINAL available-choice factor16 := mul(101370917007360000, 4)
操作後
Selection original full ordered ORIGINAL available-choice factor16 = 405483668029440000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S017
操作前
Selection original full ordered ORIGINAL available-choice factor17 := mul(405483668029440000, 3)
操作後
Selection original full ordered ORIGINAL available-choice factor17 = 1216451004088320000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S018
操作前
Selection original full ordered ORIGINAL available-choice factor18 := mul(1216451004088320000, 2)
操作後
Selection original full ordered ORIGINAL available-choice factor18 = 2432902008176640000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S019
操作前
source roles = [Selection original full ordered, 20, 19]
操作後
derived = [[20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], 2432902008176640000]
條件：無
第j位置有n-j個未用相異標記物件，完整乘k個choices；k0只有一個空ordered tuple。這不是可重複或環形／不可區分模型。
規則：Selection_ORDERED_distinct_without_replacement_available_choices · 依賴：G001, S020
操作前
Selection original unordered ordered ORIGINAL available-choice factor0 := mul(1, 20)
操作後
Selection original unordered ordered ORIGINAL available-choice factor0 = 20
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S021
操作前
Selection original unordered ordered ORIGINAL available-choice factor1 := mul(20, 19)
操作後
Selection original unordered ordered ORIGINAL available-choice factor1 = 380
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S022
操作前
Selection original unordered ordered ORIGINAL available-choice factor2 := mul(380, 18)
操作後
Selection original unordered ordered ORIGINAL available-choice factor2 = 6840
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S023
操作前
Selection original unordered ordered ORIGINAL available-choice factor3 := mul(6840, 17)
操作後
Selection original unordered ordered ORIGINAL available-choice factor3 = 116280
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S024
操作前
Selection original unordered ordered ORIGINAL available-choice factor4 := mul(116280, 16)
操作後
Selection original unordered ordered ORIGINAL available-choice factor4 = 1860480
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S025
操作前
Selection original unordered ordered ORIGINAL available-choice factor5 := mul(1860480, 15)
操作後
Selection original unordered ordered ORIGINAL available-choice factor5 = 27907200
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S026
操作前
Selection original unordered ordered ORIGINAL available-choice factor6 := mul(27907200, 14)
操作後
Selection original unordered ordered ORIGINAL available-choice factor6 = 390700800
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S027
操作前
Selection original unordered ordered ORIGINAL available-choice factor7 := mul(390700800, 13)
操作後
Selection original unordered ordered ORIGINAL available-choice factor7 = 5079110400
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S028
操作前
Selection original unordered ordered ORIGINAL available-choice factor8 := mul(5079110400, 12)
操作後
Selection original unordered ordered ORIGINAL available-choice factor8 = 60949324800
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S029
操作前
Selection original unordered ordered ORIGINAL available-choice factor9 := mul(60949324800, 11)
操作後
Selection original unordered ordered ORIGINAL available-choice factor9 = 670442572800
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S030
操作前
Selection original unordered ordered ORIGINAL available-choice factor10 := mul(670442572800, 10)
操作後
Selection original unordered ordered ORIGINAL available-choice factor10 = 6704425728000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S031
操作前
Selection original unordered ordered ORIGINAL available-choice factor11 := mul(6704425728000, 9)
操作後
Selection original unordered ordered ORIGINAL available-choice factor11 = 60339831552000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S032
操作前
Selection original unordered ordered ORIGINAL available-choice factor12 := mul(60339831552000, 8)
操作後
Selection original unordered ordered ORIGINAL available-choice factor12 = 482718652416000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S033
操作前
Selection original unordered ordered ORIGINAL available-choice factor13 := mul(482718652416000, 7)
操作後
Selection original unordered ordered ORIGINAL available-choice factor13 = 3379030566912000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S034
操作前
Selection original unordered ordered ORIGINAL available-choice factor14 := mul(3379030566912000, 6)
操作後
Selection original unordered ordered ORIGINAL available-choice factor14 = 20274183401472000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S035
操作前
Selection original unordered ordered ORIGINAL available-choice factor15 := mul(20274183401472000, 5)
操作後
Selection original unordered ordered ORIGINAL available-choice factor15 = 101370917007360000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S036
操作前
Selection original unordered ordered ORIGINAL available-choice factor16 := mul(101370917007360000, 4)
操作後
Selection original unordered ordered ORIGINAL available-choice factor16 = 405483668029440000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S037
操作前
Selection original unordered ordered ORIGINAL available-choice factor17 := mul(405483668029440000, 3)
操作後
Selection original unordered ordered ORIGINAL available-choice factor17 = 1216451004088320000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S038
操作前
Selection original unordered ordered ORIGINAL available-choice factor18 := mul(1216451004088320000, 2)
操作後
Selection original unordered ordered ORIGINAL available-choice factor18 = 2432902008176640000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S039
操作前
source roles = [Selection original unordered ordered, 20, 19]
操作後
derived = [[20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], 2432902008176640000]
條件：無
第j位置有n-j個未用相異標記物件，完整乘k個choices；k0只有一個空ordered tuple。這不是可重複或環形／不可區分模型。
規則：Selection_ORDERED_distinct_without_replacement_available_choices · 依賴：G001, S040
操作前
Selection original unordered within-subset ORIGINAL factorial factor2 := mul(1, 2)
操作後
Selection original unordered within-subset ORIGINAL factorial factor2 = 2
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S041
操作前
Selection original unordered within-subset ORIGINAL factorial factor3 := mul(2, 3)
操作後
Selection original unordered within-subset ORIGINAL factorial factor3 = 6
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S042
操作前
Selection original unordered within-subset ORIGINAL factorial factor4 := mul(6, 4)
操作後
Selection original unordered within-subset ORIGINAL factorial factor4 = 24
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S043
操作前
Selection original unordered within-subset ORIGINAL factorial factor5 := mul(24, 5)
操作後
Selection original unordered within-subset ORIGINAL factorial factor5 = 120
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S044
操作前
Selection original unordered within-subset ORIGINAL factorial factor6 := mul(120, 6)
操作後
Selection original unordered within-subset ORIGINAL factorial factor6 = 720
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S045
操作前
Selection original unordered within-subset ORIGINAL factorial factor7 := mul(720, 7)
操作後
Selection original unordered within-subset ORIGINAL factorial factor7 = 5040
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S046
操作前
Selection original unordered within-subset ORIGINAL factorial factor8 := mul(5040, 8)
操作後
Selection original unordered within-subset ORIGINAL factorial factor8 = 40320
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S047
操作前
Selection original unordered within-subset ORIGINAL factorial factor9 := mul(40320, 9)
操作後
Selection original unordered within-subset ORIGINAL factorial factor9 = 362880
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S048
操作前
Selection original unordered within-subset ORIGINAL factorial factor10 := mul(362880, 10)
操作後
Selection original unordered within-subset ORIGINAL factorial factor10 = 3628800
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S049
操作前
Selection original unordered within-subset ORIGINAL factorial factor11 := mul(3628800, 11)
操作後
Selection original unordered within-subset ORIGINAL factorial factor11 = 39916800
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S050
操作前
Selection original unordered within-subset ORIGINAL factorial factor12 := mul(39916800, 12)
操作後
Selection original unordered within-subset ORIGINAL factorial factor12 = 479001600
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S051
操作前
Selection original unordered within-subset ORIGINAL factorial factor13 := mul(479001600, 13)
操作後
Selection original unordered within-subset ORIGINAL factorial factor13 = 6227020800
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S052
操作前
Selection original unordered within-subset ORIGINAL factorial factor14 := mul(6227020800, 14)
操作後
Selection original unordered within-subset ORIGINAL factorial factor14 = 87178291200
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S053
操作前
Selection original unordered within-subset ORIGINAL factorial factor15 := mul(87178291200, 15)
操作後
Selection original unordered within-subset ORIGINAL factorial factor15 = 1307674368000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S054
操作前
Selection original unordered within-subset ORIGINAL factorial factor16 := mul(1307674368000, 16)
操作後
Selection original unordered within-subset ORIGINAL factorial factor16 = 20922789888000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S055
操作前
Selection original unordered within-subset ORIGINAL factorial factor17 := mul(20922789888000, 17)
操作後
Selection original unordered within-subset ORIGINAL factorial factor17 = 355687428096000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S056
操作前
Selection original unordered within-subset ORIGINAL factorial factor18 := mul(355687428096000, 18)
操作後
Selection original unordered within-subset ORIGINAL factorial factor18 = 6402373705728000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S057
操作前
Selection original unordered within-subset ORIGINAL factorial factor19 := mul(6402373705728000, 19)
操作後
Selection original unordered within-subset ORIGINAL factorial factor19 = 121645100408832000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S058
操作前
source roles = [Selection original unordered within-subset, 19]
操作後
derived = [121645100408832000]
條件：無
完整乘2..n，n0或1是空積1；精確BigInt不作浮點、也不把衍生count截SAFE。
規則：Selection_factorial_COMPLETE_integer_product_EMPTY0factorial1 · 依賴：G001, S059
操作前
Selection original unordered ORIGINAL exact ordered over k factorial := div(2432902008176640000, 121645100408832000)
操作後
Selection original unordered ORIGINAL exact ordered over k factorial = 20
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S060
操作前
Selection original unordered reverse unordered times k factorial := mul(20, 121645100408832000)
操作後
Selection original unordered reverse unordered times k factorial = 2432902008176640000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S061
操作前
source roles = [Selection original unordered, 20, 19, 2432902008176640000, 121645100408832000]
操作後
derived = [20, 2432902008176640000]
條件：無
每個k-subset有恰k!個orderings，所以P/k!整除為C；反乘回P驗證。空subset與空ordering各一種，不自行改成概率模型。
規則：Selection_UNORDERED_each_subset_has_kfactorial_orderings · 依賴：G001, S062
操作前
Selection original ordering factor ORIGINAL factorial factor2 := mul(1, 2)
操作後
Selection original ordering factor ORIGINAL factorial factor2 = 2
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S063
操作前
Selection original ordering factor ORIGINAL factorial factor3 := mul(2, 3)
操作後
Selection original ordering factor ORIGINAL factorial factor3 = 6
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S064
操作前
Selection original ordering factor ORIGINAL factorial factor4 := mul(6, 4)
操作後
Selection original ordering factor ORIGINAL factorial factor4 = 24
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S065
操作前
Selection original ordering factor ORIGINAL factorial factor5 := mul(24, 5)
操作後
Selection original ordering factor ORIGINAL factorial factor5 = 120
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S066
操作前
Selection original ordering factor ORIGINAL factorial factor6 := mul(120, 6)
操作後
Selection original ordering factor ORIGINAL factorial factor6 = 720
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S067
操作前
Selection original ordering factor ORIGINAL factorial factor7 := mul(720, 7)
操作後
Selection original ordering factor ORIGINAL factorial factor7 = 5040
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S068
操作前
Selection original ordering factor ORIGINAL factorial factor8 := mul(5040, 8)
操作後
Selection original ordering factor ORIGINAL factorial factor8 = 40320
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S069
操作前
Selection original ordering factor ORIGINAL factorial factor9 := mul(40320, 9)
操作後
Selection original ordering factor ORIGINAL factorial factor9 = 362880
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S070
操作前
Selection original ordering factor ORIGINAL factorial factor10 := mul(362880, 10)
操作後
Selection original ordering factor ORIGINAL factorial factor10 = 3628800
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S071
操作前
Selection original ordering factor ORIGINAL factorial factor11 := mul(3628800, 11)
操作後
Selection original ordering factor ORIGINAL factorial factor11 = 39916800
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S072
操作前
Selection original ordering factor ORIGINAL factorial factor12 := mul(39916800, 12)
操作後
Selection original ordering factor ORIGINAL factorial factor12 = 479001600
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S073
操作前
Selection original ordering factor ORIGINAL factorial factor13 := mul(479001600, 13)
操作後
Selection original ordering factor ORIGINAL factorial factor13 = 6227020800
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S074
操作前
Selection original ordering factor ORIGINAL factorial factor14 := mul(6227020800, 14)
操作後
Selection original ordering factor ORIGINAL factorial factor14 = 87178291200
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S075
操作前
Selection original ordering factor ORIGINAL factorial factor15 := mul(87178291200, 15)
操作後
Selection original ordering factor ORIGINAL factorial factor15 = 1307674368000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S076
操作前
Selection original ordering factor ORIGINAL factorial factor16 := mul(1307674368000, 16)
操作後
Selection original ordering factor ORIGINAL factorial factor16 = 20922789888000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S077
操作前
Selection original ordering factor ORIGINAL factorial factor17 := mul(20922789888000, 17)
操作後
Selection original ordering factor ORIGINAL factorial factor17 = 355687428096000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S078
操作前
Selection original ordering factor ORIGINAL factorial factor18 := mul(355687428096000, 18)
操作後
Selection original ordering factor ORIGINAL factorial factor18 = 6402373705728000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S079
操作前
Selection original ordering factor ORIGINAL factorial factor19 := mul(6402373705728000, 19)
操作後
Selection original ordering factor ORIGINAL factorial factor19 = 121645100408832000
條件：無
保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。
規則：exact_rational_arithmetic · 依賴：G001, S080
操作前
source roles = [Selection original ordering factor, 19]
操作後
derived = [121645100408832000]
條件：無
完整乘2..n，n0或1是空積1；精確BigInt不作浮點、也不把衍生count截SAFE。
規則：Selection_factorial_COMPLETE_integer_product_EMPTY0factorial1 · 依賴：G001, S081
S082 · 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。S081 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor19，不以小數近似替代。S080 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor18，不以小數近似替代。S079 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor17，不以小數近似替代。S078 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor16，不以小數近似替代。S077 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor15，不以小數近似替代。S076 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor14，不以小數近似替代。S075 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor13，不以小數近似替代。S074 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor12，不以小數近似替代。S073 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor11，不以小數近似替代。S072 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor10，不以小數近似替代。S071 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor9，不以小數近似替代。S070 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor8，不以小數近似替代。S069 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor7，不以小數近似替代。S068 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor6，不以小數近似替代。S067 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor5，不以小數近似替代。S066 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor4，不以小數近似替代。S065 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor3，不以小數近似替代。S064 · premise_and_result_audit — 以精確有理數重算 Selection original ordering factor ORIGINAL factorial factor2，不以小數近似替代。S063 · 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。S062 · premise_and_result_audit — 以精確有理數重算 Selection original unordered reverse unordered times k factorial，不以小數近似替代。S061 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ORIGINAL exact ordered over k factorial，不以小數近似替代。S060 · 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。S059 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor19，不以小數近似替代。S058 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor18，不以小數近似替代。S057 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor17，不以小數近似替代。S056 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor16，不以小數近似替代。S055 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor15，不以小數近似替代。S054 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor14，不以小數近似替代。S053 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor13，不以小數近似替代。S052 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor12，不以小數近似替代。S051 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor11，不以小數近似替代。S050 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor10，不以小數近似替代。S049 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor9，不以小數近似替代。S048 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor8，不以小數近似替代。S047 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor7，不以小數近似替代。S046 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor6，不以小數近似替代。S045 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor5，不以小數近似替代。S044 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor4，不以小數近似替代。S043 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor3，不以小數近似替代。S042 · premise_and_result_audit — 以精確有理數重算 Selection original unordered within-subset ORIGINAL factorial factor2，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 original unordered ordered ORIGINAL available-choice factor18，不以小數近似替代。S039 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor17，不以小數近似替代。S038 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor16，不以小數近似替代。S037 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor15，不以小數近似替代。S036 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor14，不以小數近似替代。S035 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor13，不以小數近似替代。S034 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor12，不以小數近似替代。S033 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor11，不以小數近似替代。S032 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor10，不以小數近似替代。S031 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor9，不以小數近似替代。S030 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor8，不以小數近似替代。S029 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor7，不以小數近似替代。S028 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor6，不以小數近似替代。S027 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor5，不以小數近似替代。S026 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor4，不以小數近似替代。S025 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor3，不以小數近似替代。S024 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor2，不以小數近似替代。S023 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor1，不以小數近似替代。S022 · premise_and_result_audit — 以精確有理數重算 Selection original unordered ordered ORIGINAL available-choice factor0，不以小數近似替代。S021 · 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。S020 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor18，不以小數近似替代。S019 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor17，不以小數近似替代。S018 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor16，不以小數近似替代。S017 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor15，不以小數近似替代。S016 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor14，不以小數近似替代。S015 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor13，不以小數近似替代。S014 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor12，不以小數近似替代。S013 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor11，不以小數近似替代。S012 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor10，不以小數近似替代。S011 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor9，不以小數近似替代。S010 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor8，不以小數近似替代。S009 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor7，不以小數近似替代。S008 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor6，不以小數近似替代。S007 · premise_and_result_audit — 以精確有理數重算 Selection original full ordered ORIGINAL available-choice factor5，不以小數近似替代。S006 · premise_and_result_audit — 以精確有理數重算 Selection original full 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 original full 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 original full 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 original full 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 original full 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。(2432902008176640000, 20, 121645100408832000, 否)
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。
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。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: ea448859106ceebfb01660a22971587895598f3783231ef5f2ce61042e07b2d8candidate record SHA-256: d3f081b08e97d22b6d0b9e7bad9fbf0384e71e0b9dd160ef68f5f2b42f657518semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bfquestion-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4Lean: NOT_RUN · human review: NOT_PERFORMED