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R0005-03020 · q-20260930-r06-combination-exact-count-0020
Select k=3 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,3).
G001 — Select k=3 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,3).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=8;k=3;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(8, 0)
After
choices at position1 = 8
Conditions: none
After i distinct selections,n−i objects remain available.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
ordered prefix count1 := mul(1, 8)
After
ordered prefix count1 = 8
Conditions: none
The product rule accumulates the ordered count without repetition.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
choices at position2 := sub(8, 1)
After
choices at position2 = 7
Conditions: none
After i distinct selections,n−i objects remain available.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
ordered prefix count2 := mul(8, 7)
After
ordered prefix count2 = 56
Conditions: none
The product rule accumulates the ordered count without repetition.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
choices at position3 := sub(8, 2)
After
choices at position3 = 6
Conditions: none
After i distinct selections,n−i objects remain available.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
ordered prefix count3 := mul(56, 6)
After
ordered prefix count3 = 336
Conditions: none
The product rule accumulates the ordered count without repetition.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
factorial through2 := mul(1, 2)
After
factorial through2 = 2
Conditions: none
Accumulate k!,the number of orders of the same selected k-element set.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
factorial through3 := mul(2, 3)
After
factorial through3 = 6
Conditions: none
Accumulate k!,the number of orders of the same selected k-element set.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
ordered selections=336;orders per subset=6
After
C(8,3)=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, S009
Before
unordered subset count := div(336, 6)
After
unordered subset count = 56
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, S010
Before
C(8,3)=n!/(k!·(n−k)!)
After
C(8,3)=40320/(6·120)=56
Conditions: none
Rewrite the falling product as n!/(n−k)! and audit the binomial formula,including boundary cases.
Rule: binomial_count_formula · Uses: G001, S011
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Recompute unordered subset count with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Recompute factorial through3 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute factorial through2 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute ordered prefix count3 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute choices at position3 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.56
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: e780797cbc947d3fcd6eb8a6cbc47ef30549171b2b046fc5a8ec79a70b548208candidate record SHA-256: 4b5b5c2349ec96c3f269ec532b9bc44181555d5a15ab1b6e7f47cf3788b6e6cfsemantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9Lean: NOT_RUN · human review: NOT_PERFORMED