Say you have a language exercise where you have to match $n>2$ words with $n$ images. If you draw $n$ edges uniformly at random step by step (first edge, starting from vertex $1$ goes to image $l_1\in\{1,\dots,n\} $ w.p. $1/n$, second edge starting at vertex $2$ goes to $l_2\in\{1,\dots,n\}\backslash \{l_1\}$ w.p. ${1\over n-1}$ etc), forming a random perfect matching after $n$ time steps, what is the expected number of correct edges?
Let $k_i$ be the number of matchings on $n-i+n-i$ vertices such that no edge is correct. (From the hat check problem).
So $P[\text{exactly }i\text{ correct edges}] = {n\choose i}\cdot {1\over n}\cdots{1\over n-i+1}\cdot k_i\cdot {1\over n-i }\cdots{1\over 2}$
And from this we can compute the expectation. Does this seem correct?