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R0005-03005 · q-20260930-r06-combination-exact-count-0005
Select k=1 objects without order, replacement, or repetition from n=3 distinct objects. The empty selection counts once. Find C(3,1).
G001 — Select k=1 objects without order, replacement, or repetition from n=3 distinct objects. The empty selection counts once. Find C(3,1).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.
Before
n=3;k=1;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.
Rule: unordered_selection_model · Uses: G001
Before
choices at position1 := sub(3, 0)
After
choices at position1 = 3
Conditions: none
After i distinct selections,n−i objects remain available.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
ordered prefix count1 := mul(1, 3)
After
ordered prefix count1 = 3
Conditions: none
The product rule accumulates the ordered count without repetition.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
ordered selections=3;orders per subset=1
After
C(3,1)=ordered selections/k!
Conditions: none
Every k-subset has exactly k! orderings,and different subsets have disjoint orderings;divide by this common positive factor.
Rule: uniform_order_fibers · Uses: G001, S003
Before
unordered subset count := div(3, 1)
After
unordered subset count = 3
Conditions: none
Exact division gives an integer;when k=0 both empty products are1 and there is exactly one empty selection.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
C(3,1)=n!/(k!·(n−k)!)
After
C(3,1)=6/(1·2)=3
Conditions: none
Rewrite the falling product as n!/(n−k)! and audit the binomial formula,including boundary cases.
Rule: binomial_count_formula · Uses: G001, S005
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Recompute unordered subset count with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.3
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: a3023480ac573ad5ecc2c6efc05b4eb67a012b81990cd0c42737b7ea0db5e9afcandidate record SHA-256: 959b07fbbd7329f54ac20f5f73da4bb44b40c94683cd61b112eb81bcf803e7casemantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9Lean: NOT_RUN · human review: NOT_PERFORMED