Multinomial Theorem

Theorem

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

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, the multinomial coefficient is defined by:


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

The multinomial theorem is the generalization of the binomial theorem.

Proof
The proof proceeds by proving with mathematical induction on $m$, the statement:


 * $\displaystyle \forall n \in \N: \left({x_1 + x_2 + \cdots + x_m}\right)^n = \sum_{k_1+k_2+\cdots+k_m \mathop = n} \binom n {k_1, k_2, \ldots, k_m} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m}$

Base Case ($m = 1$)
Trivially, for all $n \in \N$:


 * $\displaystyle \left({x_1}\right)^n = \sum_{k_1 \mathop = n} \frac{n!}{k_1!} x_1^{k_1} = \frac{n!}{n!} x_1^n = x_1^n $

Induction Hypothesis
As induction hypothesis, we use:


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

Induction Step
Now,

Therefore, using the induction hypothesis, we have shown, for any $n \in \N$:


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

thus completing the induction step.

The result follows by mathematical induction.