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Question 3025: Exact Unordered Selection Count

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

R0005-03025 · q-20260930-r06-combination-exact-count-0025

Original question and givens

Select k=6 objects without order, replacement, or repetition from n=12 distinct objects. The empty selection counts once. Find C(12,6).

  • G001 — Select k=6 objects without order, replacement, or repetition from n=12 distinct objects. The empty selection counts once. Find C(12,6).

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

n=12;k=6;distinct objects,without replacement

After

unordered k-subsets of an n-element set

Conditions: none

Each object is selected at most once and order is ignored;the ordered-selection count is not the requested answer.

Step identifiers and dependencies

Rule: unordered_selection_model · Uses: G001

S002 · calculation

Before

choices at position1 := sub(12, 0)

After

choices at position1 = 12

Conditions: none

After i distinct selections,n−i objects remain available.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

ordered prefix count1 := mul(1, 12)

After

ordered prefix count1 = 12

Conditions: none

The product rule accumulates the ordered count without repetition.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

choices at position2 := sub(12, 1)

After

choices at position2 = 11

Conditions: none

After i distinct selections,n−i objects remain available.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

ordered prefix count2 := mul(12, 11)

After

ordered prefix count2 = 132

Conditions: none

The product rule accumulates the ordered count without repetition.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

choices at position3 := sub(12, 2)

After

choices at position3 = 10

Conditions: none

After i distinct selections,n−i objects remain available.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

ordered prefix count3 := mul(132, 10)

After

ordered prefix count3 = 1320

Conditions: none

The product rule accumulates the ordered count without repetition.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

choices at position4 := sub(12, 3)

After

choices at position4 = 9

Conditions: none

After i distinct selections,n−i objects remain available.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

ordered prefix count4 := mul(1320, 9)

After

ordered prefix count4 = 11880

Conditions: none

The product rule accumulates the ordered count without repetition.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

choices at position5 := sub(12, 4)

After

choices at position5 = 8

Conditions: none

After i distinct selections,n−i objects remain available.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

ordered prefix count5 := mul(11880, 8)

After

ordered prefix count5 = 95040

Conditions: none

The product rule accumulates the ordered count without repetition.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

choices at position6 := sub(12, 5)

After

choices at position6 = 7

Conditions: none

After i distinct selections,n−i objects remain available.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

ordered prefix count6 := mul(95040, 7)

After

ordered prefix count6 = 665280

Conditions: none

The product rule accumulates the ordered count without repetition.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

factorial through2 := mul(1, 2)

After

factorial through2 = 2

Conditions: none

Accumulate k!,the number of orders of the same selected k-element set.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

factorial through3 := mul(2, 3)

After

factorial through3 = 6

Conditions: none

Accumulate k!,the number of orders of the same selected k-element set.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

factorial through4 := mul(6, 4)

After

factorial through4 = 24

Conditions: none

Accumulate k!,the number of orders of the same selected k-element set.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

factorial through5 := mul(24, 5)

After

factorial through5 = 120

Conditions: none

Accumulate k!,the number of orders of the same selected k-element set.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

factorial through6 := mul(120, 6)

After

factorial through6 = 720

Conditions: none

Accumulate k!,the number of orders of the same selected k-element set.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

ordered selections=665280;orders per subset=720

After

C(12,6)=ordered selections/k!

Conditions: none

Every k-subset has exactly k! orderings,and different subsets have disjoint orderings;divide by this common positive factor.

Step identifiers and dependencies

Rule: uniform_order_fibers · Uses: G001, S018

S020 · calculation

Before

unordered subset count := div(665280, 720)

After

unordered subset count = 924

Conditions: none

Exact division gives an integer;when k=0 both empty products are1 and there is exactly one empty selection.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

C(12,6)=n!/(k!·(n−k)!)

After

C(12,6)=479001600/(720·720)=924

Conditions: none

Rewrite the falling product as n!/(n−k)! and audit the binomial formula,including boundary cases.

Step identifiers and dependencies

Rule: binomial_count_formula · Uses: G001, S020

Reverse checks

  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Recompute unordered subset count with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Recompute factorial through6 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute factorial through5 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute factorial through4 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute factorial through3 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute factorial through2 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute ordered prefix count6 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute choices at position6 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute ordered prefix count5 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute choices at position5 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute ordered prefix count4 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute choices at position4 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute ordered prefix count3 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute choices at position3 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute ordered prefix count2 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute choices at position2 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute ordered prefix count1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute choices at position1 with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

924

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: 884d61ddb435444adef96c08cd6ac567b1e76fcb62cab093385226faff628c1c
  • candidate record SHA-256: 702aea75e52ac3996fd653ad0578af65b6be33953e0cfaab66067fc27d7801a9
  • semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2
  • question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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