Binomial theorem

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

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 — Binomial coefficients count choices.
• Mathematical induction — A general-n proof has an induction obligation.
• Univariate polynomials over the rationals — 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.