Question 413: Exact Binomial Coefficients Keep k and n Minus k Distinct

Original question and givens

Use integer n=12 with 0≤n≤50 and integer k=12 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).

• G001 — Use integer n=12 with 0≤n≤50 and integer k=12 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).

Definition and scope

Integer counting with 0≤k≤n. C(n,k) counts k-position subsets; the complement bijection gives symmetry.

• mko-binomial-theorem

Step-by-step derivation

n=12≥0; multiplication commutes

group the finite product by the number of y choices

choose 12 positions out of 12

C(12,12)=479001600/(479001600·1)=1

choose 0 positions out of 12

C(12,0)=479001600/(1·479001600)=1

choose subset of size 12

its complement has size 0

C(12,12)=1; C(12,0)=1

symmetric coefficients equal: Yes

Reverse checks

• S005 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S004 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S003 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S002 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S001 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.

Result

(12, 12, 1, 1, Yes)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

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.