Multinomial Theorem

Theorem

 * $$(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} $$

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:
 * $${n \choose k_1, k_2, \ldots, k_m} = \frac{n!}{k_1!k_2!\ldots k_m!}$$.

The multinomial theorem is the generalization of the binomial theorem.

Proof
Strong mathematical induction.

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

Inductive Hypothesis

 * $$(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} $$ for all $$ m\geq1 $$

Inductive Step
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Now,

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Therefore:


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