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 · implies
Before
source roles = [Selection all subsets ordered, 1, 0]
After
derived = [[], 1]
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, S001
S003 · implies
Before
source roles = [Selection all subsets within-subset, 0]
After
derived = [1]
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, S002
S004 · calculation
Before
Selection all subsets ORIGINAL exact ordered over k factorial := div(1, 1)
After
Selection all subsets ORIGINAL exact ordered over k factorial = 1
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 reverse unordered times k factorial := mul(1, 1)
After
Selection all subsets reverse unordered times k factorial = 1
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 · implies
Before
source roles = [Selection all subsets, 1, 0, 1, 1]
After
derived = [1, 1]
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, S005
S007 · implies
Before
source roles = [Selection subsets INCLUDING label0, 0, -1]
After
derived = [0]
Conditions: none
This is a DERIVED subset size in the partition formula.No subsets of that size exist,so the count is0;do not misclassify derived k-1=-1 as an invalid raw input.
Step identifiers and dependencies
Rule: Selection_DERIVED_binomial_outside_size_ZERO_not_input_rejection · Uses: G001, S006
S008 · implies
Before
source roles = [Selection subsets EXCLUDING label0 ordered, 0, 0]
After
derived = [[], 1]
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, S007
S009 · implies
Before
source roles = [Selection subsets EXCLUDING label0 within-subset, 0]
After
derived = [1]
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, S008
S010 · calculation
Before
Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial := div(1, 1)
After
Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial = 1
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 subsets EXCLUDING label0 reverse unordered times k factorial := mul(1, 1)
After
Selection subsets EXCLUDING label0 reverse unordered times k factorial = 1
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 subsets EXCLUDING label0, 0, 0, 1, 1]
After
derived = [1, 1]
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, S011
S013 · calculation
Before
Selection ORIGINAL included plus excluded := sum(0, 1)
After
Selection ORIGINAL included plus excluded = 1
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 partition sum minus whole := sub(1, 1)
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, S013
S015 · implies
Before
source roles = [1, 0]
After
derived = [1, 0, 1, 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, S014