(This is a cross-post from Math StackExchange https://math.stackexchange.com/questions/609641/multinomial-distribution-sum-of-squared-probabilities)

Let $\vec X = (X_1, \dots, X_k)$ be a draw from a fair multinomial distribution with $n$ trials, i.e. $P(X_1 = x_1, \dots, X_k = x_k) = \binom{n}{x_1, \dots, x_k} k^{-n}$

Let $\vec Y$ be an independent draw from the same distribution, i.e. $\vec X$ and $\vec Y$ follow the same law.

My questions is this: what is the probability that $\vec X = \vec Y$?

It appears that this probability, for fixed $k$, is roughly $(c_k + o(1)) n^{-(k-1)/2}$. This makes sense; the first $k-1$ coordinates vary on the order of $\sqrt{n}$ and are "independent-ish". (The last coordinate is of course fixed by the first $k-1$). Can one compute $c_k$?