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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]

1.Proof: Proof by Induction (related to "Multinomial Theorem")

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[1209] Matoušek, J; Nešetřil, J: “Invitation to Discrete Mathematics”, Oxford University Press, 1998