Expansion of $ \left(\sum_{i=1}^n x_i\right)^m$

Question

What is the expansion of $ \left(\sum_{i=1}^n x_i\right)^m? $

For example, \begin{align*} \left(\sum_{i=1}^n x_i\right)^2&=\sum_{i=1}^n x_i^2+2\sum_{i < j}^n x_ix_j,\\ \left(\sum_{i=1}^n x_i\right)^3&=\sum_{i=1}^n x_i^3+3\sum_{i < j}^n x_i^2x_j +6\sum_{i < j < k}^n x_ix_jx_k. \end{align*} I suspect this is some well-known result but I could not find it. (sorry if the answer is just a link to a webpage)

Answer 1

$\displaystyle \big( \sum_{i=1}^{m} x_{i} \big)^{n} = \sum_{k_{1}+k_{2}+ \cdots + k_{n}=n} \binom{n}{k_{1}, k_{2}, \ldots, k_{n}} \displaystyle \Pi_{t=1}^{m}x_{t}^{k_{t}}$, where $\binom{n}{k_{1}, k_{2}, \ldots, k_{n}} = \frac{n!} {k_{1}! k_{2}! \cdots k_{n}!} $