Fix some positive integers $N$ and $d_k$, $k=1,2,\dots$ with $N=\sum_{k=1}^\infty d_k$.

Suppose you have a graph $G$ taken randomly uniformly among the set of all (unoriented) graphs with $N$ vertices, $d_k$ of which have degree $k$ (i.e., they are connected by an edge to $k$ other vertices).

Now suppose that you (uniformly) randomly pick $M$ vertices of $G$ to form a new graph $U$, $M < N$. Join two vertices of $U$ by an edge if there is an edge in $G$ between these two vertices. Hence $U$ is a random subgraph of $G$.

Now let $p_k$ be the number of vertices in $U$ that have degree $k$.

If $M$, $N$, and $d_k$ are fixed, what is the expectation of $p_k$?