Why are there $\binom{2k+n-1}{2k}$ ways to create a homogeneous polynomial term of degree $2k$ with $n$ variables?

Question

Why are there $\binom{2k+n-1}{2k}$ ways to create a homogenious polynomial term, which is of the form:

$x_1^{a_1} + x_2^{a_2} + \dots x_n^{a_n} $ where $a_1 + a_2 + \dots + a_n = 2k$

We're looking into the set $H_{2k,n}$, the set of homogeneous polynomials of degree $2k$ with $n$ variables, and the dimension of this set is apparently $\binom{2k+n-1}{2k}$, but I can't wrap my head around it.

Anyone a clear explanation?

Answer 1

Stars and bars. You have $2 k$ exponents to distribute among $n$ variables. Think of it as $2 k$ stars separated into $n$ buckets (the variables) by $n - 1$ bars. Something like "$****|*|**||*$" ($a^4 b^1 c^2 d^0 e^1$). So you have $2 k + n - 1$ positions in all, of which you have to select $n - 1$ for the bars.