# Binomial theorem

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

With commutative multiplication, a binomial power of natural exponent n has a complete finite expansion with binomial coefficients.

mko-binomial-theorem · theorem · v0.10.0



## Mathematical statement

For n∈N_0, (x+y)^n=sum over k=0..n of C(n,k)*x^(n−k)*y^k. Coefficients are exact, all indices and both endpoint terms are retained, and the exponents must not be interchanged. At n=0 the polynomial is constant1. A combinatorial witness chooses k y-factors from n factors. Negative/noninteger exponents and noncommuting variables are outside this finite-polynomial slice.





## Representative formula (exact TeX)

(x+y)^n





## Assumptions


- n is a nonnegative safe integer, and integer k lies between0 and n.
- Evaluation companions use exact integers.





## Symbol scope


- x — first commuting term
- y — second commuting term
- n — nonnegative integer exponent





## Intuition, definition, and misconceptions

Include every k from0 through n.

Coefficient symmetry and the factorial formula are exact.

Keep the n=0 case separate from out-of-domain exponent boundaries.

Do not exchange the k and n−k exponents, omit endpoints or use this finite expansion for a negative exponent.





## Adopted derivation guide

[A full binomial expansion by choosing y factors](https://ocme.evemisslab.com/en/derivations/derivation-mko-binomial-theorem-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Permutations and combinations](https://ocme.evemisslab.com/en/mko/mko-permutation-combination/) — Binomial coefficients count choices.
- [Mathematical induction](https://ocme.evemisslab.com/en/mko/mko-mathematical-induction/) — A general-n proof has an induction obligation.
- [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/mko/mko-polynomial/) — The expansion has commutative multiplication.





## Verification boundary

Do not apply the formula unconditionally to noncommuting variables.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add Catalan numbers and generating functions.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/binomial-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-binomial-theorem/index.md)