Let $G\subset K_n$ be a subgraph of the complete graph and let $v(G)$ and $e(G)$ denote its vertex and edge set, respectively. Consider the Bernoulli graph $\mathcal G = \mathcal G(|v(G)|, p)$ on $|v(G)|$ many vertices and with edge probability $p$.
What is the expected number of copies of $G$ in $\mathcal G$?
To obtain an induced copy of $G$ we need to successfully pick $|e(G)|$ particular edges and fail to pick the remaining ${|v(G)| \choose 2} - |e(G)|$ ones. So the expectation must look something like:
$$M \cdot p^{|e(G)|} \cdot (1-p)^{{|v(G)| \choose 2} - |e(G)|}$$
where $M$ is the number of possible ways of arranging a copy in $G$. But what form does $M$ take exactly? I can't seem to figure it out and would appreciate any help.