Since the original hatcheck problem is really famous and many questions have been posted on the forum, I think I can safely skip the context part.
Note that, the probability of exactly one man receives the correct hat is 1/n, this can be solved by using the linearity of expectations. Moreover, no man receives correct hat is a derangement, which can be solved by inclusion-exclusion principle.
Now, to take a step further, if we are to calculate the probability of exactly r man receives the correct hat, how can we exploit the information we have already have to derive a simpler answer? Since all the materials online seems to be particularly interested in the above two cases, I feel rather confused.