Missing common-neighbour count in a binomial random graph ($\mathbb{E}X=\left(\genfrac{}{}{0ex}{}{n}{2}\right)(1-p^2{)}^{n-2}$)

Let $X$ count unordered vertex pairs with no common neighbour in a binomial random graph. Each other vertex provides the required pair of edges independently, with success probability $p^2$. A fixed pair has no common neighbour with probability $(1-p^2{)}^{n-2}$. Summing its indicators gives the displayed expectation.