# Question 34175: Quotient of Permutation and Combination Counts [Return to catalogue page 171](https://ocme.evemisslab.com/en/questions/page/171/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260930-r06-permutation-combination-foundations/#q-20260930-r06-permutation-combination-quotient-0025) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0067-34175 · q-20260930-r06-permutation-combination-quotient-0025 ## Original question and givens For selection of k=6 distinct objects without replacement from n=12, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once. - G001 — For selection of k=6 distinct objects without replacement from n=12, 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](https://ocme.evemisslab.com/en/mko/mko-permutation-combination/) ## Step-by-step derivation S001 · impliesBefore 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. Step identifiers and dependenciesRule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001 S002 · impliesBefore source roles = [12, 6] After derived = [[12, 11, 10, 9, 8, 7], 1] Conditions: none 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. Step identifiers and dependenciesRule: Selection_DISTINCT_no_replacement_order_and_EMPTY_convention · Uses: G001, S001 S003 · calculationBefore Permutation ORIGINAL available choices position1 := mul(1, 12) After Permutation ORIGINAL available choices position1 = 12 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Permutation ORIGINAL available choices position2 := mul(12, 11) After Permutation ORIGINAL available choices position2 = 132 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Permutation ORIGINAL available choices position3 := mul(132, 10) After Permutation ORIGINAL available choices position3 = 1320 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Permutation ORIGINAL available choices position4 := mul(1320, 9) After Permutation ORIGINAL available choices position4 = 11880 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Permutation ORIGINAL available choices position5 := mul(11880, 8) After Permutation ORIGINAL available choices position5 = 95040 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Permutation ORIGINAL available choices position6 := mul(95040, 7) After Permutation ORIGINAL available choices position6 = 665280 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Quotient ORIGINAL factorial factor1 := mul(1, 1) After Quotient ORIGINAL factorial factor1 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Quotient ORIGINAL factorial factor2 := mul(1, 2) After Quotient ORIGINAL factorial factor2 = 2 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Quotient ORIGINAL factorial factor3 := mul(2, 3) After Quotient ORIGINAL factorial factor3 = 6 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Quotient ORIGINAL factorial factor4 := mul(6, 4) After Quotient ORIGINAL factorial factor4 = 24 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Quotient ORIGINAL factorial factor5 := mul(24, 5) After Quotient ORIGINAL factorial factor5 = 120 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Quotient ORIGINAL factorial factor6 := mul(120, 6) After Quotient ORIGINAL factorial factor6 = 720 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Quotient ORIGINAL permutations divided by subset orderings := div(665280, 720) After Quotient ORIGINAL permutations divided by subset orderings = 924 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Quotient ORIGINAL permutations over combinations := div(665280, 924) After Quotient ORIGINAL permutations over combinations = 720 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [12, 6, 665280, 924] After derived = [720, 720] Conditions: none 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. Step identifiers and dependenciesRule: Selection_EVERY_k_subset_has_k_factorial_orderings · Uses: G001, S016 ## Reverse checks - S017 · 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. - S016 · premise_and_result_audit — Recompute Quotient ORIGINAL permutations over combinations with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Quotient ORIGINAL permutations divided by subset orderings with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor6 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Quotient ORIGINAL factorial factor1 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 720 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: a7e311700d39c89b618ce0e9db66c44238ed10ace44e48bc2e4f962f5ed33a77 - candidate record SHA-256: 02833b5eeca22e325b5aaca18147c7dce114caa161db54ec499c17f53a2bb012 - semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c - question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260930-r06-permutation-combination-quotient-0025.json)