Question 3021: Exact Unordered Selection Count

Original question and givens

Select k=8 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,8).

• G001 — Select k=8 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,8).

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

n=8;k=8;distinct objects,without replacement

unordered k-subsets of an n-element set

choices at position1 := sub(8, 0)

choices at position1 = 8

ordered prefix count1 := mul(1, 8)

ordered prefix count1 = 8

choices at position2 := sub(8, 1)

choices at position2 = 7

ordered prefix count2 := mul(8, 7)

ordered prefix count2 = 56

choices at position3 := sub(8, 2)

choices at position3 = 6

ordered prefix count3 := mul(56, 6)

ordered prefix count3 = 336

choices at position4 := sub(8, 3)

choices at position4 = 5

ordered prefix count4 := mul(336, 5)

ordered prefix count4 = 1680

choices at position5 := sub(8, 4)

choices at position5 = 4

ordered prefix count5 := mul(1680, 4)

ordered prefix count5 = 6720

choices at position6 := sub(8, 5)

choices at position6 = 3

ordered prefix count6 := mul(6720, 3)

ordered prefix count6 = 20160

choices at position7 := sub(8, 6)

choices at position7 = 2

ordered prefix count7 := mul(20160, 2)

ordered prefix count7 = 40320

choices at position8 := sub(8, 7)

choices at position8 = 1

ordered prefix count8 := mul(40320, 1)

ordered prefix count8 = 40320

factorial through2 := mul(1, 2)

factorial through2 = 2

factorial through3 := mul(2, 3)

factorial through3 = 6

factorial through4 := mul(6, 4)

factorial through4 = 24

factorial through5 := mul(24, 5)

factorial through5 = 120

factorial through6 := mul(120, 6)

factorial through6 = 720

factorial through7 := mul(720, 7)

factorial through7 = 5040

factorial through8 := mul(5040, 8)

factorial through8 = 40320

ordered selections=40320;orders per subset=40320

C(8,8)=ordered selections/k!

unordered subset count := div(40320, 40320)

unordered subset count = 1

C(8,8)=n!/(k!·(n−k)!)

C(8,8)=40320/(40320·1)=1

Reverse checks

• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Recompute unordered subset count with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Recompute factorial through8 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute factorial through7 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute factorial through6 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute factorial through5 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute factorial through4 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute factorial through3 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute factorial through2 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute ordered prefix count8 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute choices at position8 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute ordered prefix count7 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute choices at position7 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

1

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.