Question 34173: Quotient of Permutation and Combination Counts

Original question and givens

For selection of k=4 distinct objects without replacement from n=9, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

• G001 — For selection of k=4 distinct objects without replacement from n=9, 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 = [9, 4]

derived = [[9, 8, 7, 6], 1]

Permutation ORIGINAL available choices position1 := mul(1, 9)

Permutation ORIGINAL available choices position1 = 9

Permutation ORIGINAL available choices position2 := mul(9, 8)

Permutation ORIGINAL available choices position2 = 72

Permutation ORIGINAL available choices position3 := mul(72, 7)

Permutation ORIGINAL available choices position3 = 504

Permutation ORIGINAL available choices position4 := mul(504, 6)

Permutation ORIGINAL available choices position4 = 3024

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 permutations divided by subset orderings := div(3024, 24)

Quotient ORIGINAL permutations divided by subset orderings = 126

Quotient ORIGINAL permutations over combinations := div(3024, 126)

Quotient ORIGINAL permutations over combinations = 24

source roles = [9, 4, 3024, 126]

derived = [24, 24]

Reverse checks

• S013 · 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.
• S012 · premise_and_result_audit — Recompute Quotient ORIGINAL permutations over combinations with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Quotient ORIGINAL permutations divided by subset orderings with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor1 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

24

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.