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.
Proofs: 1
Parts: 1
Proofs: 2
Propositions: 3
