Say we expand $\left(\sum_{i=1}^n x_i\right)^k$ into monomials.
If $k=3$ there are $3n(n-1)$ monomials with two variables: $3x_1x_2^2 + 3x_1x_3^2 +\dots + 3x_1^2x_2 + \dots$.
Is there a closed form or asymptotic describing this number? That is, given $n$ and $k$, count the number of monomials with $a$ variables?
(Note that this is different from counting the number of monomials of a given degree, of which there is of course a simple closed form.)
This is the number of ways to select an ordered $a$-tuple of $n$ variables times the number of ways to partition $k$ objects into $a$ subsets:
$$ a!\binom na\left\{k\atop a\right\}=\binom na\sum_{j=0}^a(-1)^{a-j}\binom ajj^k\;, $$
where $\left\{k\atop a\right\}$ is a Stirling number of the second kind.