For selection of k=1 distinct objects without replacement from n=3, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.
G001 — For selection of k=1 distinct objects without replacement from n=3, 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.
source roles = [qf-permutation-combination-quotient, [k, n, operation]]
After
derived = [quotient, no-invented-domain]
Conditions: none
Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations.
Choose ordered positions from n DISTINCT objects WITHOUT replacement.Position i has n−i+1 choices.For k=0,the empty selection counts ONCE and the product starts at1.THIS bank permits n≤200,not another bank’s32/64 cap.
Permutation ORIGINAL available choices position1 := mul(1, 3)
After
Permutation ORIGINAL available choices position1 = 3
Conditions: none
Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.
Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.
Quotient ORIGINAL permutations divided by subset orderings := div(3, 1)
After
Quotient ORIGINAL permutations divided by subset orderings = 3
Conditions: none
Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.
Quotient ORIGINAL permutations over combinations := div(3, 3)
After
Quotient ORIGINAL permutations over combinations = 1
Conditions: none
Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.
EVERY k-element subset has exactly k! orderings.Ordered selections partition into equal fibers over their underlying subsets,so P=C*k!,C>0,and P/C=k!.At k=0 this remains1/1=1,not an estimated ratio or probability.
S007 · 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.
S006 · premise_and_result_audit — Recompute Quotient ORIGINAL permutations over combinations with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute Quotient ORIGINAL permutations divided by subset orderings with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor1 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
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.