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Theorem: Multinomial Theorem

Let $$n\in\mathbb N$$ be a natural number and let $$x_1,x_2,\ldots,x_m$$ be some complex or real numbers. Then the sum with $$r$$ terms raised to the $$n$$-th power of will expand as follows:

$(x_1 + x_2 + \ldots + x_r)^n=\sum_{\substack{k_1+\ldots+k_r=n \\ k_1,\ldots,k_r}}\binom{n}{k_1,k_2\ldots,k_r}x_1^{k_1} x_2^{k_2}\ldots x_r^{k_r}.$

where

$\binom n{k_1,k_2,\ldots,k_m}:=\frac{n !}{k_1 !k_2 !\cdot\ldots \cdot k_m !}$
is the multinomial coefficient.

| | | | | created: 2016-03-25 10:16:00 | modified: 2019-09-01 14:39:38 | by: bookofproofs | references: [1209]

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