Question 30897: Combinations Including or Excluding a Fixed Existing Label

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0060-30897 · q-20261011-advance-20261015-r06-permutation-combination-0072

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=22≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

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=22≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

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.

mko-permutation-combination

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, 22)

After

Selection all subsets ordered ORIGINAL available-choice factor0 = 22

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(22, 21)

After

Selection all subsets ordered ORIGINAL available-choice factor1 = 462

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(462, 20)

After

Selection all subsets ordered ORIGINAL available-choice factor2 = 9240

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 · implies

Before

source roles = [Selection all subsets ordered, 22, 3]

After

derived = [[22, 21, 20], 9240]

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, S004

S006 · 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, S005

S007 · 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, S006

S008 · implies

Before

source roles = [Selection all subsets within-subset, 3]

After

derived = [6]

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, S007

S009 · calculation

Before

Selection all subsets ORIGINAL exact ordered over k factorial := div(9240, 6)

After

Selection all subsets ORIGINAL exact ordered over k factorial = 1540

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 reverse unordered times k factorial := mul(1540, 6)

After

Selection all subsets reverse unordered times k factorial = 9240

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 · implies

Before

source roles = [Selection all subsets, 22, 3, 9240, 6]

After

derived = [1540, 9240]

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, S010

S012 · calculation

Before

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 21)

After

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 = 21

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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(21, 20)

After

Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 = 420

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 · implies

Before

source roles = [Selection subsets INCLUDING label0 ordered, 21, 2]

After

derived = [[21, 20], 420]

Conditions: none

Step identifiers and dependencies

Rule: Selection_ORDERED_distinct_without_replacement_available_choices · Uses: G001, S013

S015 · 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, S014

S016 · implies

Before

source roles = [Selection subsets INCLUDING label0 within-subset, 2]

After

derived = [2]

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, S015

S017 · calculation

Before

Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial := div(420, 2)

After

Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial = 210

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 subsets INCLUDING label0 reverse unordered times k factorial := mul(210, 2)

After

Selection subsets INCLUDING label0 reverse unordered times k factorial = 420

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 · implies

Before

source roles = [Selection subsets INCLUDING label0, 21, 2, 420, 2]

After

derived = [210, 420]

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, S018

S020 · calculation

Before

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 21)

After

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 = 21

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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(21, 20)

After

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 = 420

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 · calculation

Before

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 := mul(420, 19)

After

Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 = 7980

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 · implies

Before

source roles = [Selection subsets EXCLUDING label0 ordered, 21, 3]

After

derived = [[21, 20, 19], 7980]

Conditions: none

Step identifiers and dependencies

Rule: Selection_ORDERED_distinct_without_replacement_available_choices · Uses: G001, S022

S024 · 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, S023

S025 · 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, S024

S026 · implies

Before

source roles = [Selection subsets EXCLUDING label0 within-subset, 3]

After

derived = [6]

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, S025

S027 · calculation

Before

Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial := div(7980, 6)

After

Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial = 1330

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 EXCLUDING label0 reverse unordered times k factorial := mul(1330, 6)

After

Selection subsets EXCLUDING label0 reverse unordered times k factorial = 7980

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 · implies

Before

source roles = [Selection subsets EXCLUDING label0, 21, 3, 7980, 6]

After

derived = [1330, 7980]

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, S028

S030 · calculation

Before

Selection ORIGINAL included plus excluded := sum(210, 1330)

After

Selection ORIGINAL included plus excluded = 1540

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 partition sum minus whole := sub(1540, 1540)

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, S030

S032 · implies

Before

source roles = [22, 3]

After

derived = [1540, 210, 1330, 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, S031

Reverse checks

S032 · 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.
S031 · premise_and_result_audit — Recompute Selection ORIGINAL partition sum minus whole with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute Selection ORIGINAL included plus excluded with exact rational arithmetic, not rounded decimals.
S029 · 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.
S028 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 reverse unordered times k factorial with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial with exact rational arithmetic, not rounded decimals.
S026 · 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.
S025 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
S023 · 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.
S022 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals.
S019 · 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.
S018 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 reverse unordered times k factorial with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial with exact rational arithmetic, not rounded decimals.
S016 · 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.
S015 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
S014 · 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.
S013 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals.
S011 · 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.
S010 · premise_and_result_audit — Recompute Selection all subsets reverse unordered times k factorial with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute Selection all subsets ORIGINAL exact ordered over k factorial with exact rational arithmetic, not rounded decimals.
S008 · 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.
S007 · premise_and_result_audit — Recompute Selection all subsets within-subset ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute Selection all subsets within-subset ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
S005 · 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.
S004 · premise_and_result_audit — Recompute Selection all subsets ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute Selection all subsets ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute Selection all subsets 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

(1540, 210, 1330, 0)

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: bb2ea6ddb84cc6ba728be531406149358f9fdafe9875c67a0cce6ba2957cca43
candidate record SHA-256: 55e3d06072c5e312238442177010a0dbe14af998eeb948c2182c3b6667fe84c9
semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf
question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON