# Question 407: Exact Binomial Coefficients Keep k and n Minus k Distinct

[Return to catalogue page 3](https://ocme.evemisslab.com/en/questions/page/3/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261109-r05-binomial-theorem-foundations/#q-20261014-advance-20261109-r05-binomial-theorem-0007)

This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

P1000-0407 · q-20261014-advance-20261109-r05-binomial-theorem-0007



## Original question and givens

Use integer n=6 with 0≤n≤50 and integer k=6 with 0≤k≤n. The coefficient setting has commutative multiplication. Give (n, k, coefficient, symmetric_coefficient, equal).


- G001 — Use integer n=6 with 0≤n≤50 and integer k=6 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](https://ocme.evemisslab.com/en/mko/mko-binomial-theorem/)





## Step-by-step derivation

S001 · impliesBefore

n=6≥0; multiplication commutes

After

group the finite product by the number of y choices

Conditions: none

Choose x or y in each factor and group by the number of y choices; count coefficients as integers first.

Step identifiers and dependenciesRule: commutative_product · Uses: G001





S002 · impliesBefore

choose 6 positions out of 6

After

C(6,6)=720/(720·1)=1

Conditions: none

Count each k-position subset once, dividing out reorderings inside each of the two groups.

Step identifiers and dependenciesRule: subset_count · Uses: G001, S001





S003 · impliesBefore

choose 0 positions out of 6

After

C(6,0)=720/(1·720)=1

Conditions: none

Count each k-position subset once, dividing out reorderings inside each of the two groups.

Step identifiers and dependenciesRule: subset_count · Uses: G001, S002





S004 · impliesBefore

choose subset of size 6

After

its complement has size 0

Conditions: none

Taking complements is an invertible map, so the two counts agree.

Step identifiers and dependenciesRule: subset_complement_bijection · Uses: G001, S003





S005 · impliesBefore

C(6,6)=1; C(6,0)=1

After

symmetric coefficients equal: Yes

Conditions: none

The values agree with the complement bijection; n=0 still has exactly the empty subset.

Step identifiers and dependenciesRule: coefficient_symmetry · Uses: G001, S004









## 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

(6, 6, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 57516a5baac20b64c71536209bd84481b5f1e96ca6a28d7422b7a2287b2f6e7e
- candidate record SHA-256: 66ae48eeecfe7802782730bb588895e78ddfc5eb53f147f0ecc39ea6df03cb9b
- semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
- question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261109-r05-binomial-theorem-0007.json)