Let $N \gg k\geq 1$, and $\gamma \in (0,1/2)$. I have a set of $N$ colored balls (3 colors in total), where $N$ is a multiple of 

* $\gamma N$ are red;
* $\gamma N$ are blue;
* $(1-2\gamma)N$ are green.

I pick a random subset $S$ of $[N]$ of size $k$, and would like to know the expectation and variance of the quantity
$$
(X-Y)^2
$$
where $X$ (resp. $Y$) is the number of red (resp. blue) balls. *(Ideally, the full distribution of $X-Y$ would be nice, but the above is all I really need, I think).*

Now, this does feel an awful lot like a variant of balls-and-bins, or basically of something I should know. But every attempt I have made today either relies on (a) the (not very justified) approximation of independence (i.e., sampling with replacement instead of without); or (b) horrible looking sums involving binomial coefficients.

Does the problem above have a name, or a long history that I am missing because of not knowing the right keywords? Or, if not, is there a very simple analysis of the two quantities mentioned above?