Multinomial Theorem

Theorem

 * $\displaystyle (x_1 + x_2 + \cdots + x_m)^n =  \sum_{k_1+k_2\ldots+k_m=n} \binom n {k_1, k_2, \ldots, k_m}

x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m} $

where $m$ is a positive integer and $n$ is non-negative.

The sum is taken for all non-negative integers $k_1, k_2, \ldots, k_m$ such that $k_1 + k_2 + \cdots + k_m = n$.

Also:
 * $\displaystyle \binom n {k_1, k_2, \ldots, k_m} = \frac{n!}{k_1!k_2!\ldots k_m!}$

The multinomial theorem is the generalization of the binomial theorem.

Proof
Proof by strong mathematical induction:

Base Case (m=1)
Trivial:
 * $\displaystyle (x_1)^n = \sum_{k_1=n} \frac{n!}{k_1!} x_1^{k_1} = \frac{n!}{n!} x_1^n = x_1^n $

Induction Hypothesis

 * $\displaystyle \forall m \in \N, m \ge 1: (x_1 + x_2 + \cdots + x_m)^n = \sum_{k_1+k_2\ldots+k_m=n} {n \choose k_1, k_2, \ldots, k_m} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m}$

Induction Step
Now,

Therefore:


 * $\displaystyle (x_1 + x_2 + \cdots + x_m + x_{m+1})^n = \sum_{k_1 + k_2 + \cdots + k_m + k_{m+1} = n} \binom n {k_1, k_2, \ldots, k_m, k_{m+1}} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m} x_{m+1}^{k_{m+1}}$