Step-by-step derivation
S001 · implies
Before
source roles = [qf-density15-selection-one-distinguished-counts, density15_selection_one_distinguished_counts, finite_distinct_selection_protocol]
After
derived = [[total, included_label0, excluded_label0, partition_defect]]
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 all subsets ordered ORIGINAL available-choice factor0 := mul(1, 20)
After
Selection all subsets 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 all subsets ordered ORIGINAL available-choice factor1 := mul(20, 19)
After
Selection all subsets 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 all subsets ordered ORIGINAL available-choice factor2 := mul(380, 18)
After
Selection all subsets 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 all subsets ordered ORIGINAL available-choice factor3 := mul(6840, 17)
After
Selection all subsets 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 all subsets ordered ORIGINAL available-choice factor4 := mul(116280, 16)
After
Selection all subsets 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 all subsets ordered ORIGINAL available-choice factor5 := mul(1860480, 15)
After
Selection all subsets 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 all subsets ordered ORIGINAL available-choice factor6 := mul(27907200, 14)
After
Selection all subsets 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 all subsets ordered ORIGINAL available-choice factor7 := mul(390700800, 13)
After
Selection all subsets 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 all subsets ordered ORIGINAL available-choice factor8 := mul(5079110400, 12)
After
Selection all subsets 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 all subsets ordered ORIGINAL available-choice factor9 := mul(60949324800, 11)
After
Selection all subsets 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 · implies
Before
source roles = [Selection all subsets ordered, 20, 10]
After
derived = [[20, 19, 18, 17, 16, 15, 14, 13, 12, 11], 670442572800]
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, S011
S013 · calculation
Before
Selection all subsets within-subset ORIGINAL factorial factor2 := mul(1, 2)
After
Selection all subsets 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, S012
S014 · calculation
Before
Selection all subsets within-subset ORIGINAL factorial factor3 := mul(2, 3)
After
Selection all subsets 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, S013
S015 · calculation
Before
Selection all subsets within-subset ORIGINAL factorial factor4 := mul(6, 4)
After
Selection all subsets 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, S014
S016 · calculation
Before
Selection all subsets within-subset ORIGINAL factorial factor5 := mul(24, 5)
After
Selection all subsets 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, S015
S017 · calculation
Before
Selection all subsets within-subset ORIGINAL factorial factor6 := mul(120, 6)
After
Selection all subsets 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, S016
S018 · calculation
Before
Selection all subsets within-subset ORIGINAL factorial factor7 := mul(720, 7)
After
Selection all subsets 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, S017
S019 · calculation
Before
Selection all subsets within-subset ORIGINAL factorial factor8 := mul(5040, 8)
After
Selection all subsets 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, S018
S020 · calculation
Before
Selection all subsets within-subset ORIGINAL factorial factor9 := mul(40320, 9)
After
Selection all subsets 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, S019
S021 · calculation
Before
Selection all subsets within-subset ORIGINAL factorial factor10 := mul(362880, 10)
After
Selection all subsets 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, S020
S022 · implies
Before
source roles = [Selection all subsets within-subset, 10]
After
derived = [3628800]
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, S021
S023 · calculation
Before
Selection all subsets ORIGINAL exact ordered over k factorial := div(670442572800, 3628800)
After
Selection all subsets ORIGINAL exact ordered over k factorial = 184756
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 all subsets reverse unordered times k factorial := mul(184756, 3628800)
After
Selection all subsets reverse unordered times k factorial = 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, S023
S025 · implies
Before
source roles = [Selection all subsets, 20, 10, 670442572800, 3628800]
After
derived = [184756, 670442572800]
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, S024
S026 · calculation
Before
Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 19)
After
Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 = 19
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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(19, 18)
After
Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 = 342
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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor2 := mul(342, 17)
After
Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor2 = 5814
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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor3 := mul(5814, 16)
After
Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor3 = 93024
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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor4 := mul(93024, 15)
After
Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor4 = 1395360
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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor5 := mul(1395360, 14)
After
Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor5 = 19535040
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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor6 := mul(19535040, 13)
After
Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor6 = 253955520
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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor7 := mul(253955520, 12)
After
Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor7 = 3047466240
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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor8 := mul(3047466240, 11)
After
Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor8 = 33522128640
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 · implies
Before
source roles = [Selection subsets INCLUDING label0 ordered, 19, 9]
After
derived = [[19, 18, 17, 16, 15, 14, 13, 12, 11], 33522128640]
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, S034
S036 · calculation
Before
Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor2 := mul(1, 2)
After
Selection subsets INCLUDING label0 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, S035
S037 · calculation
Before
Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor3 := mul(2, 3)
After
Selection subsets INCLUDING label0 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, S036
S038 · calculation
Before
Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor4 := mul(6, 4)
After
Selection subsets INCLUDING label0 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, S037
S039 · calculation
Before
Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor5 := mul(24, 5)
After
Selection subsets INCLUDING label0 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, S038
S040 · calculation
Before
Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor6 := mul(120, 6)
After
Selection subsets INCLUDING label0 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, S039
S041 · calculation
Before
Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor7 := mul(720, 7)
After
Selection subsets INCLUDING label0 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, S040
S042 · calculation
Before
Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor8 := mul(5040, 8)
After
Selection subsets INCLUDING label0 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, S041
S043 · calculation
Before
Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor9 := mul(40320, 9)
After
Selection subsets INCLUDING label0 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, S042
S044 · implies
Before
source roles = [Selection subsets INCLUDING label0 within-subset, 9]
After
derived = [362880]
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, S043
S045 · calculation
Before
Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial := div(33522128640, 362880)
After
Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial = 92378
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 subsets INCLUDING label0 reverse unordered times k factorial := mul(92378, 362880)
After
Selection subsets INCLUDING label0 reverse unordered times k factorial = 33522128640
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 · implies
Before
source roles = [Selection subsets INCLUDING label0, 19, 9, 33522128640, 362880]
After
derived = [92378, 33522128640]
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, S046
S048 · calculation
Before
Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 19)
After
Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 = 19
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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(19, 18)
After
Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 = 342
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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 := mul(342, 17)
After
Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 = 5814
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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor3 := mul(5814, 16)
After
Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor3 = 93024
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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor4 := mul(93024, 15)
After
Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor4 = 1395360
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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor5 := mul(1395360, 14)
After
Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor5 = 19535040
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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor6 := mul(19535040, 13)
After
Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor6 = 253955520
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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor7 := mul(253955520, 12)
After
Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor7 = 3047466240
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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor8 := mul(3047466240, 11)
After
Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor8 = 33522128640
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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor9 := mul(33522128640, 10)
After
Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor9 = 335221286400
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 · implies
Before
source roles = [Selection subsets EXCLUDING label0 ordered, 19, 10]
After
derived = [[19, 18, 17, 16, 15, 14, 13, 12, 11, 10], 335221286400]
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, S057
S059 · calculation
Before
Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor2 := mul(1, 2)
After
Selection subsets EXCLUDING label0 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, S058
S060 · calculation
Before
Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor3 := mul(2, 3)
After
Selection subsets EXCLUDING label0 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, S059
S061 · calculation
Before
Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor4 := mul(6, 4)
After
Selection subsets EXCLUDING label0 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, S060
S062 · calculation
Before
Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor5 := mul(24, 5)
After
Selection subsets EXCLUDING label0 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, S061
S063 · calculation
Before
Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor6 := mul(120, 6)
After
Selection subsets EXCLUDING label0 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, S062
S064 · calculation
Before
Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor7 := mul(720, 7)
After
Selection subsets EXCLUDING label0 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, S063
S065 · calculation
Before
Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor8 := mul(5040, 8)
After
Selection subsets EXCLUDING label0 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, S064
S066 · calculation
Before
Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor9 := mul(40320, 9)
After
Selection subsets EXCLUDING label0 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, S065
S067 · calculation
Before
Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor10 := mul(362880, 10)
After
Selection subsets EXCLUDING label0 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, S066
S068 · implies
Before
source roles = [Selection subsets EXCLUDING label0 within-subset, 10]
After
derived = [3628800]
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, S067
S069 · calculation
Before
Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial := div(335221286400, 3628800)
After
Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial = 92378
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 subsets EXCLUDING label0 reverse unordered times k factorial := mul(92378, 3628800)
After
Selection subsets EXCLUDING label0 reverse unordered times k factorial = 335221286400
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 · implies
Before
source roles = [Selection subsets EXCLUDING label0, 19, 10, 335221286400, 3628800]
After
derived = [92378, 335221286400]
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, S070
S072 · calculation
Before
Selection ORIGINAL included plus excluded := sum(92378, 92378)
After
Selection ORIGINAL included plus excluded = 184756
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 partition sum minus whole := sub(184756, 184756)
After
Selection ORIGINAL partition sum minus whole = 0
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 · implies
Before
source roles = [20, 10]
After
derived = [184756, 92378, 92378, 0]
Conditions: none
n>=1 ensures the GIVEN existing label0.Containing it leaves k-1 choices from the other n-1 objects;excluding it chooses k from those n-1.The two classes are disjoint and exhaustive.At k0 inclusion count0,and at k=n exclusion count0 are legal.
Step identifiers and dependencies
Rule: Selection_FIXED_EXISTING_label0_disjoint_complete_subset_partition · Uses: G001, S073