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R0005-03024 · q-20260930-r06-combination-exact-count-0024
Select k=5 objects without order, replacement, or repetition from n=10 distinct objects. The empty selection counts once. Find C(10,5).
G001 — Select k=5 objects without order, replacement, or repetition from n=10 distinct objects. The empty selection counts once. Find C(10,5).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=10;k=5;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(10, 0)
After
choices at position1 = 10
Conditions: none
After i distinct selections,n−i objects remain available.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
ordered prefix count1 := mul(1, 10)
After
ordered prefix count1 = 10
Conditions: none
The product rule accumulates the ordered count without repetition.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
choices at position2 := sub(10, 1)
After
choices at position2 = 9
Conditions: none
After i distinct selections,n−i objects remain available.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
ordered prefix count2 := mul(10, 9)
After
ordered prefix count2 = 90
Conditions: none
The product rule accumulates the ordered count without repetition.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
choices at position3 := sub(10, 2)
After
choices at position3 = 8
Conditions: none
After i distinct selections,n−i objects remain available.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
ordered prefix count3 := mul(90, 8)
After
ordered prefix count3 = 720
Conditions: none
The product rule accumulates the ordered count without repetition.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
choices at position4 := sub(10, 3)
After
choices at position4 = 7
Conditions: none
After i distinct selections,n−i objects remain available.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
ordered prefix count4 := mul(720, 7)
After
ordered prefix count4 = 5040
Conditions: none
The product rule accumulates the ordered count without repetition.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
choices at position5 := sub(10, 4)
After
choices at position5 = 6
Conditions: none
After i distinct selections,n−i objects remain available.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
ordered prefix count5 := mul(5040, 6)
After
ordered prefix count5 = 30240
Conditions: none
The product rule accumulates the ordered count without repetition.
Rule: exact_rational_arithmetic · Uses: G001, S010
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, S011
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, S012
Before
factorial through4 := mul(6, 4)
After
factorial through4 = 24
Conditions: none
Accumulate k!,the number of orders of the same selected k-element set.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
factorial through5 := mul(24, 5)
After
factorial through5 = 120
Conditions: none
Accumulate k!,the number of orders of the same selected k-element set.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
ordered selections=30240;orders per subset=120
After
C(10,5)=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, S015
Before
unordered subset count := div(30240, 120)
After
unordered subset count = 252
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, S016
Before
C(10,5)=n!/(k!·(n−k)!)
After
C(10,5)=3628800/(120·120)=252
Conditions: none
Rewrite the falling product as n!/(n−k)! and audit the binomial formula,including boundary cases.
Rule: binomial_count_formula · Uses: G001, S017
S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Recompute unordered subset count with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Recompute factorial through5 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute factorial through4 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute factorial through3 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute factorial through2 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute ordered prefix count5 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute choices at position5 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute ordered prefix count4 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute choices at position4 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.252
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.
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