# Question 3011: Exact Unordered Selection Count [Return to catalogue page 16](https://ocme.evemisslab.com/en/questions/page/16/) [← 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-combination-exact-count-0011) 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. R0005-03011 · q-20260930-r06-combination-exact-count-0011 ## Original question and givens Select k=2 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,2). - G001 — Select k=2 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,2). ## 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 n=5;k=2;distinct objects,without replacement After unordered k-subsets of an n-element set Conditions: none Each object is selected at most once and order is ignored;the ordered-selection count is not the requested answer. Step identifiers and dependenciesRule: unordered_selection_model · Uses: G001 S002 · calculationBefore choices at position1 := sub(5, 0) After choices at position1 = 5 Conditions: none After i distinct selections,n−i objects remain available. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore ordered prefix count1 := mul(1, 5) After ordered prefix count1 = 5 Conditions: none The product rule accumulates the ordered count without repetition. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore choices at position2 := sub(5, 1) After choices at position2 = 4 Conditions: none After i distinct selections,n−i objects remain available. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore ordered prefix count2 := mul(5, 4) After ordered prefix count2 = 20 Conditions: none The product rule accumulates the ordered count without repetition. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore factorial through2 := mul(1, 2) After factorial through2 = 2 Conditions: none Accumulate k!,the number of orders of the same selected k-element set. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore ordered selections=20;orders per subset=2 After C(5,2)=ordered selections/k! Conditions: none Every k-subset has exactly k! orderings,and different subsets have disjoint orderings;divide by this common positive factor. Step identifiers and dependenciesRule: uniform_order_fibers · Uses: G001, S006 S008 · calculationBefore unordered subset count := div(20, 2) After unordered subset count = 10 Conditions: none Exact division gives an integer;when k=0 both empty products are1 and there is exactly one empty selection. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore C(5,2)=n!/(k!·(n−k)!) After C(5,2)=120/(2·6)=10 Conditions: none Rewrite the falling product as n!/(n−k)! and audit the binomial formula,including boundary cases. Step identifiers and dependenciesRule: binomial_count_formula · Uses: G001, S008 ## Reverse checks - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute unordered subset count with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute factorial through2 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute ordered prefix count2 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute choices at position2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute ordered prefix count1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute choices at position1 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result 10 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: 3fab4ae6489843a35d9715dda9903be03bf1ad582bce949b1be8bebd50682c01 - candidate record SHA-256: 180169a42505ad48794bb4802a649f834efb5285ffa0befddbff93b41f43a33c - semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2 - question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260930-r06-combination-exact-count-0011.json)