Binomial Theorem/Integral Index

Theorem
Let $X$ be one of the standard number systems $\N$, $\Z$, $\Q$, $\R$ or $\C$.

Let $x, y \in X$.

Then:

where $\dbinom n k$ is $n$ choose $k$.

Basis for the Induction
For $n = 0$ we have:


 * $\ds \paren {x + y}^0 = 1 = \binom 0 0 x^{0 - 0} y^0 = \sum_{k \mathop = 0}^0 \binom 0 k x^{0 - k} y^k$

This is the basis for the induction.

Induction Hypothesis
This is our induction hypothesis:


 * $\ds \paren {x + y}^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$

Induction Step
This is our induction step:

The result follows by the Principle of Mathematical Induction.

Also see

 * Definition:Binomial Coefficient