Question: A group of n people play "Secret Santa" as follows: each puts his or her name on a slip of paper in a hat, picks a name randomly from the hat (without replacement), and then buys a gift for that person. Unfortunately, they overlook the possibility of drawing one's own name, so some may have to buy gifts for themselves (on the bright side, some may like self-selected gifts better). Assume n ≥ 2.
(a) Find the expected number of people who pick their own names.
(b) Find the expected number of pairs of people, A and B, such that A picks B's name and B picks A's name (where A ≠ B and order doesn't matter).