R0060-30854 · q-20261011-advance-20261015-r06-permutation-combination-0079
相異標記物件無放回，有序與無序分開。n=32、k=32、r=16、model=distinct_without_replacement，前r個位置固定原序標記=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。
G001 — 相異標記物件無放回，有序與無序分開。n=32、k=32、r=16、model=distinct_without_replacement，前r個位置固定原序標記=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]。求 (剩餘物件數, 剩餘位置數, 完成數, 全有序數, 完成數/全數)。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-forced-initial-order, density15_selection_forced_initial_order, finite_distinct_selection_protocol]
操作後
derived = [[remaining_objects, remaining_positions, completions, full_ordered_count, COUNT_ratio]]
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 prefix ORIGINAL n minus used r := sub(32, 16)
操作後
Selection prefix ORIGINAL n minus used r = 16
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 prefix ORIGINAL k minus filled r := sub(32, 16)
操作後
Selection prefix ORIGINAL k minus filled r = 16
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 prefix remaining ordered ORIGINAL available-choice factor0 := mul(1, 16)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor0 = 16
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 prefix remaining ordered ORIGINAL available-choice factor1 := mul(16, 15)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor1 = 240
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 prefix remaining ordered ORIGINAL available-choice factor2 := mul(240, 14)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor2 = 3360
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 prefix remaining ordered ORIGINAL available-choice factor3 := mul(3360, 13)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor3 = 43680
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 prefix remaining ordered ORIGINAL available-choice factor4 := mul(43680, 12)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor4 = 524160
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 prefix remaining ordered ORIGINAL available-choice factor5 := mul(524160, 11)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor5 = 5765760
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 prefix remaining ordered ORIGINAL available-choice factor6 := mul(5765760, 10)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor6 = 57657600
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 prefix remaining ordered ORIGINAL available-choice factor7 := mul(57657600, 9)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor7 = 518918400
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 prefix remaining ordered ORIGINAL available-choice factor8 := mul(518918400, 8)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor8 = 4151347200
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 prefix remaining ordered ORIGINAL available-choice factor9 := mul(4151347200, 7)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor9 = 29059430400
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 prefix remaining ordered ORIGINAL available-choice factor10 := mul(29059430400, 6)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor10 = 174356582400
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 prefix remaining ordered ORIGINAL available-choice factor11 := mul(174356582400, 5)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor11 = 871782912000
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 prefix remaining ordered ORIGINAL available-choice factor12 := mul(871782912000, 4)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor12 = 3487131648000
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 prefix remaining ordered ORIGINAL available-choice factor13 := mul(3487131648000, 3)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor13 = 10461394944000
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 prefix remaining ordered ORIGINAL available-choice factor14 := mul(10461394944000, 2)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor14 = 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, S017
操作前
Selection prefix remaining ordered ORIGINAL available-choice factor15 := mul(20922789888000, 1)
操作後
Selection prefix remaining ordered ORIGINAL available-choice factor15 = 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, S018
操作前
source roles = [Selection prefix remaining ordered, 16, 16]
操作後
derived = [[16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], 20922789888000]
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, S019
操作前
Selection full ordered ORIGINAL available-choice factor0 := mul(1, 32)
操作後
Selection full ordered ORIGINAL available-choice factor0 = 32
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, S020
操作前
Selection full ordered ORIGINAL available-choice factor1 := mul(32, 31)
操作後
Selection full ordered ORIGINAL available-choice factor1 = 992
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 full ordered ORIGINAL available-choice factor2 := mul(992, 30)
操作後
Selection full ordered ORIGINAL available-choice factor2 = 29760
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 full ordered ORIGINAL available-choice factor3 := mul(29760, 29)
操作後
Selection full ordered ORIGINAL available-choice factor3 = 863040
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 full ordered ORIGINAL available-choice factor4 := mul(863040, 28)
操作後
Selection full ordered ORIGINAL available-choice factor4 = 24165120
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 full ordered ORIGINAL available-choice factor5 := mul(24165120, 27)
操作後
Selection full ordered ORIGINAL available-choice factor5 = 652458240
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 full ordered ORIGINAL available-choice factor6 := mul(652458240, 26)
操作後
Selection full ordered ORIGINAL available-choice factor6 = 16963914240
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 full ordered ORIGINAL available-choice factor7 := mul(16963914240, 25)
操作後
Selection full ordered ORIGINAL available-choice factor7 = 424097856000
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 full ordered ORIGINAL available-choice factor8 := mul(424097856000, 24)
操作後
Selection full ordered ORIGINAL available-choice factor8 = 10178348544000
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 full ordered ORIGINAL available-choice factor9 := mul(10178348544000, 23)
操作後
Selection full ordered ORIGINAL available-choice factor9 = 234102016512000
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 full ordered ORIGINAL available-choice factor10 := mul(234102016512000, 22)
操作後
Selection full ordered ORIGINAL available-choice factor10 = 5150244363264000
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 full ordered ORIGINAL available-choice factor11 := mul(5150244363264000, 21)
操作後
Selection full ordered ORIGINAL available-choice factor11 = 108155131628544000
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 full ordered ORIGINAL available-choice factor12 := mul(108155131628544000, 20)
操作後
Selection full ordered ORIGINAL available-choice factor12 = 2163102632570880000
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 full ordered ORIGINAL available-choice factor13 := mul(2163102632570880000, 19)
操作後
Selection full ordered ORIGINAL available-choice factor13 = 41098950018846720000
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 full ordered ORIGINAL available-choice factor14 := mul(41098950018846720000, 18)
操作後
Selection full ordered ORIGINAL available-choice factor14 = 739781100339240960000
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 full ordered ORIGINAL available-choice factor15 := mul(739781100339240960000, 17)
操作後
Selection full ordered ORIGINAL available-choice factor15 = 12576278705767096320000
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 full ordered ORIGINAL available-choice factor16 := mul(12576278705767096320000, 16)
操作後
Selection full ordered ORIGINAL available-choice factor16 = 201220459292273541120000
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 full ordered ORIGINAL available-choice factor17 := mul(201220459292273541120000, 15)
操作後
Selection full ordered ORIGINAL available-choice factor17 = 3018306889384103116800000
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 full ordered ORIGINAL available-choice factor18 := mul(3018306889384103116800000, 14)
操作後
Selection full ordered ORIGINAL available-choice factor18 = 42256296451377443635200000
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 full ordered ORIGINAL available-choice factor19 := mul(42256296451377443635200000, 13)
操作後
Selection full ordered ORIGINAL available-choice factor19 = 549331853867906767257600000
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
Selection full ordered ORIGINAL available-choice factor20 := mul(549331853867906767257600000, 12)
操作後
Selection full ordered ORIGINAL available-choice factor20 = 6591982246414881207091200000
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, S040
操作前
Selection full ordered ORIGINAL available-choice factor21 := mul(6591982246414881207091200000, 11)
操作後
Selection full ordered ORIGINAL available-choice factor21 = 72511804710563693278003200000
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 full ordered ORIGINAL available-choice factor22 := mul(72511804710563693278003200000, 10)
操作後
Selection full ordered ORIGINAL available-choice factor22 = 725118047105636932780032000000
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 full ordered ORIGINAL available-choice factor23 := mul(725118047105636932780032000000, 9)
操作後
Selection full ordered ORIGINAL available-choice factor23 = 6526062423950732395020288000000
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 full ordered ORIGINAL available-choice factor24 := mul(6526062423950732395020288000000, 8)
操作後
Selection full ordered ORIGINAL available-choice factor24 = 52208499391605859160162304000000
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 full ordered ORIGINAL available-choice factor25 := mul(52208499391605859160162304000000, 7)
操作後
Selection full ordered ORIGINAL available-choice factor25 = 365459495741241014121136128000000
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 full ordered ORIGINAL available-choice factor26 := mul(365459495741241014121136128000000, 6)
操作後
Selection full ordered ORIGINAL available-choice factor26 = 2192756974447446084726816768000000
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 full ordered ORIGINAL available-choice factor27 := mul(2192756974447446084726816768000000, 5)
操作後
Selection full ordered ORIGINAL available-choice factor27 = 10963784872237230423634083840000000
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 full ordered ORIGINAL available-choice factor28 := mul(10963784872237230423634083840000000, 4)
操作後
Selection full ordered ORIGINAL available-choice factor28 = 43855139488948921694536335360000000
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 full ordered ORIGINAL available-choice factor29 := mul(43855139488948921694536335360000000, 3)
操作後
Selection full ordered ORIGINAL available-choice factor29 = 131565418466846765083609006080000000
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 full ordered ORIGINAL available-choice factor30 := mul(131565418466846765083609006080000000, 2)
操作後
Selection full ordered ORIGINAL available-choice factor30 = 263130836933693530167218012160000000
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 full ordered ORIGINAL available-choice factor31 := mul(263130836933693530167218012160000000, 1)
操作後
Selection full ordered ORIGINAL available-choice factor31 = 263130836933693530167218012160000000
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
source roles = [Selection full ordered, 32, 32]
操作後
derived = [[32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], 263130836933693530167218012160000000]
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, S052
操作前
Selection ORIGINAL completion COUNT ratio := div(20922789888000, 263130836933693530167218012160000000)
操作後
Selection ORIGINAL completion COUNT ratio = 1/12576278705767096320000
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
source roles = [32, 32, 16, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]]
操作後
derived = [16, 16, 20922789888000, 263130836933693530167218012160000000, 1/12576278705767096320000]
條件：無
前r位置固定用0..r-1且次序已給，先扣已用物件／已填位置。r0不減count，r=k只一個completion；最後是count比，沒有uniform law不可自行叫概率。
規則：Selection_FIXED_prefix_labels_and_COUNT_ratio_NOT_probability · 依賴：G001, S054
S055 · 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。S054 · premise_and_result_audit — 以精確有理數重算 Selection ORIGINAL completion COUNT ratio，不以小數近似替代。S053 · 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。S052 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor31，不以小數近似替代。S051 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor30，不以小數近似替代。S050 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor29，不以小數近似替代。S049 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor28，不以小數近似替代。S048 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor27，不以小數近似替代。S047 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor26，不以小數近似替代。S046 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor25，不以小數近似替代。S045 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor24，不以小數近似替代。S044 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor23，不以小數近似替代。S043 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor22，不以小數近似替代。S042 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor21，不以小數近似替代。S041 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor20，不以小數近似替代。S040 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor19，不以小數近似替代。S039 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor18，不以小數近似替代。S038 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor17，不以小數近似替代。S037 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor16，不以小數近似替代。S036 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor15，不以小數近似替代。S035 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor14，不以小數近似替代。S034 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor13，不以小數近似替代。S033 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor12，不以小數近似替代。S032 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor11，不以小數近似替代。S031 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor10，不以小數近似替代。S030 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor9，不以小數近似替代。S029 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor8，不以小數近似替代。S028 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor7，不以小數近似替代。S027 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor6，不以小數近似替代。S026 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor5，不以小數近似替代。S025 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor4，不以小數近似替代。S024 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor3，不以小數近似替代。S023 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor2，不以小數近似替代。S022 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor1，不以小數近似替代。S021 · premise_and_result_audit — 以精確有理數重算 Selection full ordered ORIGINAL available-choice factor0，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 prefix remaining ordered ORIGINAL available-choice factor15，不以小數近似替代。S018 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor14，不以小數近似替代。S017 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor13，不以小數近似替代。S016 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor12，不以小數近似替代。S015 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor11，不以小數近似替代。S014 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor10，不以小數近似替代。S013 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor9，不以小數近似替代。S012 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor8，不以小數近似替代。S011 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor7，不以小數近似替代。S010 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor6，不以小數近似替代。S009 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor5，不以小數近似替代。S008 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor4，不以小數近似替代。S007 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor3，不以小數近似替代。S006 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor2，不以小數近似替代。S005 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor1，不以小數近似替代。S004 · premise_and_result_audit — 以精確有理數重算 Selection prefix remaining ordered ORIGINAL available-choice factor0，不以小數近似替代。S003 · premise_and_result_audit — 以精確有理數重算 Selection prefix ORIGINAL k minus filled r，不以小數近似替代。S002 · premise_and_result_audit — 以精確有理數重算 Selection prefix ORIGINAL n minus used r，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。(16, 16, 20922789888000, 263130836933693530167218012160000000, 1/12576278705767096320000)
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: 29c987a6fe7e864ae9330dc96522934f6b1fd311f9f374009cb6911f4c745577candidate record SHA-256: 9f9ad2dd71a9686b73c95367526eeb4671ca936aec2a6b797181ed07e0468e94semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bfquestion-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4Lean: NOT_RUN · human review: NOT_PERFORMED