# Question 34196: Exact Ordered Selection Count [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-exact-count-0021) 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. R0067-34196 · q-20260930-r06-permutation-exact-count-0021 ## Original question and givens Select k=8 objects in order, without replacement or repetition, from n=8 distinct objects. The empty selection counts once. Find P(8,8). - G001 — Select k=8 objects in order, without replacement or repetition, from n=8 distinct objects. The empty selection counts once. Find P(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](https://ocme.evemisslab.com/en/mko/mko-permutation-combination/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-permutation-exact-count, [k, n, operation]] After derived = [permutation, 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 = [8, 8] After derived = [[8, 7, 6, 5, 4, 3, 2, 1], 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, 8) After Permutation ORIGINAL available choices position1 = 8 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(8, 7) After Permutation ORIGINAL available choices position2 = 56 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(56, 6) After Permutation ORIGINAL available choices position3 = 336 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(336, 5) After Permutation ORIGINAL available choices position4 = 1680 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(1680, 4) After Permutation ORIGINAL available choices position5 = 6720 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(6720, 3) After Permutation ORIGINAL available choices position6 = 20160 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 Permutation ORIGINAL available choices position7 := mul(20160, 2) After Permutation ORIGINAL available choices position7 = 40320 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 Permutation ORIGINAL available choices position8 := mul(40320, 1) After Permutation ORIGINAL available choices position8 = 40320 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 · impliesBefore source roles = [8, 8] After derived = [40320] Conditions: none The multiplication principle combines ALL k position counts,giving P(n,k)=n!/(n−k)!.This is a count,not probability.The empty product is1 and derived BigInts are not truncated to SAFE. Step identifiers and dependenciesRule: Selection_ALL_ordered_choices_falling_product · Uses: G001, S010 ## Reverse checks - S011 · 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. - 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 93ba60ceba7f65b8d5ab485b7417fdecbe986bd313a46099c80a1e82bdbe4e40 - candidate record SHA-256: ea20f3bf7ee64bfcd58b7b6051e36bfa00310eb548a92f2f32e8e0c630d54acc - 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-exact-count-0021.json)