Transpositions

Question

Let $P = (p_1,\dots, p_n)$ be a uniformly random permutation of $[n] = \{1,2,\dots,n\}$. We say $(i,j)$ is a $\textit{transposition}$ if $p_i = j$ and $p_j = i$ for $i < j$. For example, if $n = 4$, then in $(1,2,4,3)$, $(3,4)$ is a transposition, as the fourth element is $3$ and the third element is $4$. Find the expected number of transpositions in $P$ when $n = 10$.