$E[|T\cap S|^2]$ for random sets $S$, $T$ with fixed number of member $|S|=|T|=d$

Question

Let two sets $T,S\subset\{1,\dots, p\}$ be given. Both sets have $d<p$ elements, i.e. $|S|=|T|=d$

Find $E[|T\cap S|^2]$ for random sets $S$, $T$, i.e. if $\mathcal{S}$ is the set over all those sets, find

$$\frac{1}{|\mathcal{S}|^2}\sum_{S,T\in \mathcal{S}}|S\cap T|^2$$

Is this simply $$\sum_{k=0}^p \frac{\binom{d}{k}\binom{d}{d-k}}{d!}k^2=\frac{4^{d-1}d\Gamma\left(d-\frac{1}{2}\right)}{\sqrt{\pi}\left(\left(d-1\right)!\right)^2}$$ Probably this also must be dependent of $p$, doesn't it?

Answer 1

I would like to mention an alternative.

For $i=1,\dots,p$ define random variable $X_i$ as taking value $1$ if $i\in S\cap T$ and taking value $0$ otherwise.

Then:$$|S\cap T|^2=\left(\sum_{i=1}^pX_i\right)^2=\sum_{i=1}^p\sum_{j=1}^pX_iX_j$$Using symmetry we then find that:$$\mathbb E|S\cap T|^2=p\mathbb EX_1^2+p(p-1)\mathbb EX_1X_2=p\Pr(1\in S\cap T)+p(p-1)\Pr(1,2\in S\cap T)$$

Work this out...