We draw two rows of $n$ points with labels $\{ 1,2, \dots,n \}$, as shown in the figure below.
We connect each point on the top with a point on the bottom at random, making sure that no two points on the top are connected to the same point on the bottom. We say point $j$ on the top is special if it gets connected to point $j$ below.
Let $X$ be the number of special points. What is $\mathbb{E}(X)$, the expected value of $X?$
