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R0005-03001 · q-20260930-r06-combination-exact-count-0001
Select k=0 objects without order, replacement, or repetition from n=0 distinct objects. The empty selection counts once. Find C(0,0).
G001 — Select k=0 objects without order, replacement, or repetition from n=0 distinct objects. The empty selection counts once. Find C(0,0).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=0;k=0;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
ordered selections=1;orders per subset=1
After
C(0,0)=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, S001
Before
unordered subset count := div(1, 1)
After
unordered subset count = 1
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, S002
Before
C(0,0)=n!/(k!·(n−k)!)
After
C(0,0)=1/(1·1)=1
Conditions: none
Rewrite the falling product as n!/(n−k)! and audit the binomial formula,including boundary cases.
Rule: binomial_count_formula · Uses: G001, S003
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 unordered subset count with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.1
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: 8610efb84508278534ff50ee9088b898bcd66bf9f99b6eb33ac707927e6f8d43candidate record SHA-256: 0bdb532d849bc6cbaaa1ccb5893c5ab8344032ec104a979ff8476927d0baf611semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9Lean: NOT_RUN · human review: NOT_PERFORMED