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Question 30903: Ordered/Unordered Counts, Factorial, and Empty Selection

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0060-30903 · q-20261011-advance-20261015-r06-permutation-combination-0003

Original question and givens

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=20, k=19, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

  • G001 — Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=20, k=19, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-density15-selection-order-factor-triple, density15_selection_order_factor_triple, finite_distinct_selection_protocol]

After

derived = [[ordered_count, unordered_count, ordering_factor, empty_selection]]

Conditions: none

density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status.

Step identifiers and dependencies

Rule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001

S002 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor0 := mul(1, 20)

After

Selection original full ordered ORIGINAL available-choice factor0 = 20

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor1 := mul(20, 19)

After

Selection original full ordered ORIGINAL available-choice factor1 = 380

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor2 := mul(380, 18)

After

Selection original full ordered ORIGINAL available-choice factor2 = 6840

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor3 := mul(6840, 17)

After

Selection original full ordered ORIGINAL available-choice factor3 = 116280

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor4 := mul(116280, 16)

After

Selection original full ordered ORIGINAL available-choice factor4 = 1860480

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor5 := mul(1860480, 15)

After

Selection original full ordered ORIGINAL available-choice factor5 = 27907200

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor6 := mul(27907200, 14)

After

Selection original full ordered ORIGINAL available-choice factor6 = 390700800

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor7 := mul(390700800, 13)

After

Selection original full ordered ORIGINAL available-choice factor7 = 5079110400

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor8 := mul(5079110400, 12)

After

Selection original full ordered ORIGINAL available-choice factor8 = 60949324800

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor9 := mul(60949324800, 11)

After

Selection original full ordered ORIGINAL available-choice factor9 = 670442572800

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor10 := mul(670442572800, 10)

After

Selection original full ordered ORIGINAL available-choice factor10 = 6704425728000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor11 := mul(6704425728000, 9)

After

Selection original full ordered ORIGINAL available-choice factor11 = 60339831552000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor12 := mul(60339831552000, 8)

After

Selection original full ordered ORIGINAL available-choice factor12 = 482718652416000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor13 := mul(482718652416000, 7)

After

Selection original full ordered ORIGINAL available-choice factor13 = 3379030566912000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor14 := mul(3379030566912000, 6)

After

Selection original full ordered ORIGINAL available-choice factor14 = 20274183401472000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor15 := mul(20274183401472000, 5)

After

Selection original full ordered ORIGINAL available-choice factor15 = 101370917007360000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor16 := mul(101370917007360000, 4)

After

Selection original full ordered ORIGINAL available-choice factor16 = 405483668029440000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor17 := mul(405483668029440000, 3)

After

Selection original full ordered ORIGINAL available-choice factor17 = 1216451004088320000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

Selection original full ordered ORIGINAL available-choice factor18 := mul(1216451004088320000, 2)

After

Selection original full ordered ORIGINAL available-choice factor18 = 2432902008176640000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

source roles = [Selection original full ordered, 20, 19]

After

derived = [[20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], 2432902008176640000]

Conditions: none

At position j there are n-j UNUSED distinct labeled objects.Multiply ALL k available-choice counts;at k0 there is exactly ONE empty ordered tuple.This is not replacement,circular or indistinguishable selection.

Step identifiers and dependencies

Rule: Selection_ORDERED_distinct_without_replacement_available_choices · Uses: G001, S020

S022 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor0 := mul(1, 20)

After

Selection original unordered ordered ORIGINAL available-choice factor0 = 20

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor1 := mul(20, 19)

After

Selection original unordered ordered ORIGINAL available-choice factor1 = 380

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor2 := mul(380, 18)

After

Selection original unordered ordered ORIGINAL available-choice factor2 = 6840

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor3 := mul(6840, 17)

After

Selection original unordered ordered ORIGINAL available-choice factor3 = 116280

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor4 := mul(116280, 16)

After

Selection original unordered ordered ORIGINAL available-choice factor4 = 1860480

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor5 := mul(1860480, 15)

After

Selection original unordered ordered ORIGINAL available-choice factor5 = 27907200

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor6 := mul(27907200, 14)

After

Selection original unordered ordered ORIGINAL available-choice factor6 = 390700800

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor7 := mul(390700800, 13)

After

Selection original unordered ordered ORIGINAL available-choice factor7 = 5079110400

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor8 := mul(5079110400, 12)

After

Selection original unordered ordered ORIGINAL available-choice factor8 = 60949324800

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor9 := mul(60949324800, 11)

After

Selection original unordered ordered ORIGINAL available-choice factor9 = 670442572800

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor10 := mul(670442572800, 10)

After

Selection original unordered ordered ORIGINAL available-choice factor10 = 6704425728000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor11 := mul(6704425728000, 9)

After

Selection original unordered ordered ORIGINAL available-choice factor11 = 60339831552000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor12 := mul(60339831552000, 8)

After

Selection original unordered ordered ORIGINAL available-choice factor12 = 482718652416000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor13 := mul(482718652416000, 7)

After

Selection original unordered ordered ORIGINAL available-choice factor13 = 3379030566912000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor14 := mul(3379030566912000, 6)

After

Selection original unordered ordered ORIGINAL available-choice factor14 = 20274183401472000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor15 := mul(20274183401472000, 5)

After

Selection original unordered ordered ORIGINAL available-choice factor15 = 101370917007360000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor16 := mul(101370917007360000, 4)

After

Selection original unordered ordered ORIGINAL available-choice factor16 = 405483668029440000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor17 := mul(405483668029440000, 3)

After

Selection original unordered ordered ORIGINAL available-choice factor17 = 1216451004088320000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor18 := mul(1216451004088320000, 2)

After

Selection original unordered ordered ORIGINAL available-choice factor18 = 2432902008176640000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · implies

Before

source roles = [Selection original unordered ordered, 20, 19]

After

derived = [[20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], 2432902008176640000]

Conditions: none

At position j there are n-j UNUSED distinct labeled objects.Multiply ALL k available-choice counts;at k0 there is exactly ONE empty ordered tuple.This is not replacement,circular or indistinguishable selection.

Step identifiers and dependencies

Rule: Selection_ORDERED_distinct_without_replacement_available_choices · Uses: G001, S040

S042 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor2 := mul(1, 2)

After

Selection original unordered within-subset ORIGINAL factorial factor2 = 2

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor3 := mul(2, 3)

After

Selection original unordered within-subset ORIGINAL factorial factor3 = 6

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor4 := mul(6, 4)

After

Selection original unordered within-subset ORIGINAL factorial factor4 = 24

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor5 := mul(24, 5)

After

Selection original unordered within-subset ORIGINAL factorial factor5 = 120

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor6 := mul(120, 6)

After

Selection original unordered within-subset ORIGINAL factorial factor6 = 720

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor7 := mul(720, 7)

After

Selection original unordered within-subset ORIGINAL factorial factor7 = 5040

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor8 := mul(5040, 8)

After

Selection original unordered within-subset ORIGINAL factorial factor8 = 40320

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor9 := mul(40320, 9)

After

Selection original unordered within-subset ORIGINAL factorial factor9 = 362880

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor10 := mul(362880, 10)

After

Selection original unordered within-subset ORIGINAL factorial factor10 = 3628800

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor11 := mul(3628800, 11)

After

Selection original unordered within-subset ORIGINAL factorial factor11 = 39916800

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor12 := mul(39916800, 12)

After

Selection original unordered within-subset ORIGINAL factorial factor12 = 479001600

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor13 := mul(479001600, 13)

After

Selection original unordered within-subset ORIGINAL factorial factor13 = 6227020800

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor14 := mul(6227020800, 14)

After

Selection original unordered within-subset ORIGINAL factorial factor14 = 87178291200

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor15 := mul(87178291200, 15)

After

Selection original unordered within-subset ORIGINAL factorial factor15 = 1307674368000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor16 := mul(1307674368000, 16)

After

Selection original unordered within-subset ORIGINAL factorial factor16 = 20922789888000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor17 := mul(20922789888000, 17)

After

Selection original unordered within-subset ORIGINAL factorial factor17 = 355687428096000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor18 := mul(355687428096000, 18)

After

Selection original unordered within-subset ORIGINAL factorial factor18 = 6402373705728000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor19 := mul(6402373705728000, 19)

After

Selection original unordered within-subset ORIGINAL factorial factor19 = 121645100408832000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · implies

Before

source roles = [Selection original unordered within-subset, 19]

After

derived = [121645100408832000]

Conditions: none

Multiply the COMPLETE2..n factors;at n0/1 the empty product is1.Use exact BigInt,not floats or a SAFE cap on derived counts.

Step identifiers and dependencies

Rule: Selection_factorial_COMPLETE_integer_product_EMPTY0factorial1 · Uses: G001, S059

S061 · calculation

Before

Selection original unordered ORIGINAL exact ordered over k factorial := div(2432902008176640000, 121645100408832000)

After

Selection original unordered ORIGINAL exact ordered over k factorial = 20

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · calculation

Before

Selection original unordered reverse unordered times k factorial := mul(20, 121645100408832000)

After

Selection original unordered reverse unordered times k factorial = 2432902008176640000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · implies

Before

source roles = [Selection original unordered, 20, 19, 2432902008176640000, 121645100408832000]

After

derived = [20, 2432902008176640000]

Conditions: none

EACH k-subset has exactly k! orderings,so P/k! is the exact integer C.Multiply back to verify P.An empty subset and empty ordering each count once;no probability model is asserted.

Step identifiers and dependencies

Rule: Selection_UNORDERED_each_subset_has_kfactorial_orderings · Uses: G001, S062

S064 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor2 := mul(1, 2)

After

Selection original ordering factor ORIGINAL factorial factor2 = 2

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor3 := mul(2, 3)

After

Selection original ordering factor ORIGINAL factorial factor3 = 6

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor4 := mul(6, 4)

After

Selection original ordering factor ORIGINAL factorial factor4 = 24

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S065

S067 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor5 := mul(24, 5)

After

Selection original ordering factor ORIGINAL factorial factor5 = 120

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor6 := mul(120, 6)

After

Selection original ordering factor ORIGINAL factorial factor6 = 720

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor7 := mul(720, 7)

After

Selection original ordering factor ORIGINAL factorial factor7 = 5040

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor8 := mul(5040, 8)

After

Selection original ordering factor ORIGINAL factorial factor8 = 40320

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor9 := mul(40320, 9)

After

Selection original ordering factor ORIGINAL factorial factor9 = 362880

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor10 := mul(362880, 10)

After

Selection original ordering factor ORIGINAL factorial factor10 = 3628800

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor11 := mul(3628800, 11)

After

Selection original ordering factor ORIGINAL factorial factor11 = 39916800

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor12 := mul(39916800, 12)

After

Selection original ordering factor ORIGINAL factorial factor12 = 479001600

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S073

S075 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor13 := mul(479001600, 13)

After

Selection original ordering factor ORIGINAL factorial factor13 = 6227020800

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor14 := mul(6227020800, 14)

After

Selection original ordering factor ORIGINAL factorial factor14 = 87178291200

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S075

S077 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor15 := mul(87178291200, 15)

After

Selection original ordering factor ORIGINAL factorial factor15 = 1307674368000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S076

S078 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor16 := mul(1307674368000, 16)

After

Selection original ordering factor ORIGINAL factorial factor16 = 20922789888000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S077

S079 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor17 := mul(20922789888000, 17)

After

Selection original ordering factor ORIGINAL factorial factor17 = 355687428096000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor18 := mul(355687428096000, 18)

After

Selection original ordering factor ORIGINAL factorial factor18 = 6402373705728000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S079

S081 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor19 := mul(6402373705728000, 19)

After

Selection original ordering factor ORIGINAL factorial factor19 = 121645100408832000

Conditions: none

Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S080

S082 · implies

Before

source roles = [Selection original ordering factor, 19]

After

derived = [121645100408832000]

Conditions: none

Multiply the COMPLETE2..n factors;at n0/1 the empty product is1.Use exact BigInt,not floats or a SAFE cap on derived counts.

Step identifiers and dependencies

Rule: Selection_factorial_COMPLETE_integer_product_EMPTY0factorial1 · Uses: G001, S081

Reverse checks

  • S082 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
  • S081 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor19 with exact rational arithmetic, not rounded decimals.
  • S080 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor18 with exact rational arithmetic, not rounded decimals.
  • S079 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor17 with exact rational arithmetic, not rounded decimals.
  • S078 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor16 with exact rational arithmetic, not rounded decimals.
  • S077 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor15 with exact rational arithmetic, not rounded decimals.
  • S076 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor14 with exact rational arithmetic, not rounded decimals.
  • S075 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor13 with exact rational arithmetic, not rounded decimals.
  • S074 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor12 with exact rational arithmetic, not rounded decimals.
  • S073 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor11 with exact rational arithmetic, not rounded decimals.
  • S072 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor10 with exact rational arithmetic, not rounded decimals.
  • S071 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor9 with exact rational arithmetic, not rounded decimals.
  • S070 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor8 with exact rational arithmetic, not rounded decimals.
  • S069 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor7 with exact rational arithmetic, not rounded decimals.
  • S068 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor6 with exact rational arithmetic, not rounded decimals.
  • S067 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals.
  • S066 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals.
  • S065 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals.
  • S064 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
  • S063 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
  • S062 · premise_and_result_audit — Recompute Selection original unordered reverse unordered times k factorial with exact rational arithmetic, not rounded decimals.
  • S061 · premise_and_result_audit — Recompute Selection original unordered ORIGINAL exact ordered over k factorial with exact rational arithmetic, not rounded decimals.
  • S060 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
  • S059 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor19 with exact rational arithmetic, not rounded decimals.
  • S058 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor18 with exact rational arithmetic, not rounded decimals.
  • S057 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor17 with exact rational arithmetic, not rounded decimals.
  • S056 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor16 with exact rational arithmetic, not rounded decimals.
  • S055 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor15 with exact rational arithmetic, not rounded decimals.
  • S054 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor14 with exact rational arithmetic, not rounded decimals.
  • S053 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor13 with exact rational arithmetic, not rounded decimals.
  • S052 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor12 with exact rational arithmetic, not rounded decimals.
  • S051 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor11 with exact rational arithmetic, not rounded decimals.
  • S050 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor10 with exact rational arithmetic, not rounded decimals.
  • S049 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor9 with exact rational arithmetic, not rounded decimals.
  • S048 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor8 with exact rational arithmetic, not rounded decimals.
  • S047 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor7 with exact rational arithmetic, not rounded decimals.
  • S046 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor6 with exact rational arithmetic, not rounded decimals.
  • S045 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals.
  • S044 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals.
  • S043 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals.
  • S042 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
  • S041 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
  • S040 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor18 with exact rational arithmetic, not rounded decimals.
  • S039 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor17 with exact rational arithmetic, not rounded decimals.
  • S038 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor16 with exact rational arithmetic, not rounded decimals.
  • S037 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor15 with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor14 with exact rational arithmetic, not rounded decimals.
  • S035 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor13 with exact rational arithmetic, not rounded decimals.
  • S034 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor12 with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor11 with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor10 with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor9 with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor8 with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor7 with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor6 with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor5 with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor4 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor3 with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
  • S020 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor18 with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor17 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor16 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor15 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor14 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor13 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor12 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor11 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor10 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor9 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor8 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor7 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor6 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor5 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor4 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor3 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.

Result

(2432902008176640000, 20, 121645100408832000, No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: ea448859106ceebfb01660a22971587895598f3783231ef5f2ce61042e07b2d8
  • candidate record SHA-256: d3f081b08e97d22b6d0b9e7bad9fbf0384e71e0b9dd160ef68f5f2b42f657518
  • semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf
  • question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON