For a group of $130$ people, assuming that each person is equally likely to have a birthday on each of $365$ days in the year, compute a) the expected number of days of the year that are birthdays of exactly $3$ people and b) the expected number of distinct birthdays.
I can't figure out what I'm doing wrong. I let $X$ denote the number of people that share a given birthday, so $X \sim \text{Binomial}\left(130,\frac {1}{365}\right)$. I then found $$P(X=3) = \binom{130}{3}\cdot\left(\frac {1}{365}\right)^3\cdot\left(\frac {364}{365}\right)^{127}\approx0.005189.$$ I then let $Y$ denote the number of days of the year that are birthdays of $3$ people. So $Y \sim \text{Binomial}(365,0.005189)$, which gives $E(Y) = 365\cdot0.005189\approx1.89$. However this is incorrect and I can't figure out what's wrong with my approach. I approached (b) in the same way.