What is the expected number of days in a year in which exactly $k$ people in a group of $n$ people have been born?

Question

There is a group of $n$ people and we must find the average number of days that in each of them exactly $k$ people are born ($k$ and $n$ are given).

This question assumes that a year has $365$ days, and each day of the year is equally likely to be a birthday for someone.

Answer 1

By linearity of expectation, this is just $365$ times the probability that exactly $k$ people are born on a given day, which is $\binom nk(1/365)^k(364/365)^{n-k}=\binom nk364^{n-k}/365^n$, so the expected number of such days is $\binom nk364^{n-k}/365^{n-1}$.