Robust Overlap of Two Random Subsets

Let $S$ and $T$ be two subsets of $\{1, 2, \dots, n\}$, each chosen independently and uniformly at random from all $\binom{n}{k}$ subsets of size $k$ (where $1 \le k \le n$). Find the expected size of their intersection $|S \cap T|$.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.