 Solution

Let $I_{ij}$ be the indicator random variable that the unordered vertex pair $\{i,j\}$ has no common neighbour. For each of the other $n-2$ vertices, the two required edges are both present with probability $p^2$. The edge pairs for different candidate neighbours are disjoint sets of independent random choices. Thus $\mathbb{E}{I}_{ij}=(1-p^2{)}^{n-2}$. Linearity of expectation yieldsthe missing common-neighbour count in a binomial random graph.