Question 34171: Quotient of Permutation and Combination Counts

Original question and givens

For selection of k=8 distinct objects without replacement from n=8, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

• G001 — For selection of k=8 distinct objects without replacement from n=8, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

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

source roles = [qf-permutation-combination-quotient, [k, n, operation]]

derived = [quotient, no-invented-domain]

source roles = [8, 8]

derived = [[8, 7, 6, 5, 4, 3, 2, 1], 1]

Permutation ORIGINAL available choices position1 := mul(1, 8)

Permutation ORIGINAL available choices position1 = 8

Permutation ORIGINAL available choices position2 := mul(8, 7)

Permutation ORIGINAL available choices position2 = 56

Permutation ORIGINAL available choices position3 := mul(56, 6)

Permutation ORIGINAL available choices position3 = 336

Permutation ORIGINAL available choices position4 := mul(336, 5)

Permutation ORIGINAL available choices position4 = 1680

Permutation ORIGINAL available choices position5 := mul(1680, 4)

Permutation ORIGINAL available choices position5 = 6720

Permutation ORIGINAL available choices position6 := mul(6720, 3)

Permutation ORIGINAL available choices position6 = 20160

Permutation ORIGINAL available choices position7 := mul(20160, 2)

Permutation ORIGINAL available choices position7 = 40320

Permutation ORIGINAL available choices position8 := mul(40320, 1)

Permutation ORIGINAL available choices position8 = 40320

Quotient ORIGINAL factorial factor1 := mul(1, 1)

Quotient ORIGINAL factorial factor1 = 1

Quotient ORIGINAL factorial factor2 := mul(1, 2)

Quotient ORIGINAL factorial factor2 = 2

Quotient ORIGINAL factorial factor3 := mul(2, 3)

Quotient ORIGINAL factorial factor3 = 6

Quotient ORIGINAL factorial factor4 := mul(6, 4)

Quotient ORIGINAL factorial factor4 = 24

Quotient ORIGINAL factorial factor5 := mul(24, 5)

Quotient ORIGINAL factorial factor5 = 120

Quotient ORIGINAL factorial factor6 := mul(120, 6)

Quotient ORIGINAL factorial factor6 = 720

Quotient ORIGINAL factorial factor7 := mul(720, 7)

Quotient ORIGINAL factorial factor7 = 5040

Quotient ORIGINAL factorial factor8 := mul(5040, 8)

Quotient ORIGINAL factorial factor8 = 40320

Quotient ORIGINAL permutations divided by subset orderings := div(40320, 40320)

Quotient ORIGINAL permutations divided by subset orderings = 1

Quotient ORIGINAL permutations over combinations := div(40320, 1)

Quotient ORIGINAL permutations over combinations = 40320

source roles = [8, 8, 40320, 1]

derived = [40320, 40320]

Reverse checks

• S021 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
• S020 · premise_and_result_audit — Recompute Quotient ORIGINAL permutations over combinations with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Quotient ORIGINAL permutations divided by subset orderings with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor8 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor7 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor6 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position8 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position7 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position6 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position5 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position4 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position3 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
• S001 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.

Result

40320

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.