combinatorics - Probability that n people collectively occupy all 365 birthdays

The problem is quite simple to formulate. If you have a large group of people (n > 365), and their birthdays are uniformly distributed over the year (365 days), what's the probability that every day of the year is someone's birthday?

I am thinking that the problem should be equivalent to finding the number of ways to place n unlabeled balls into k labeled boxes, such that all boxes are non-empty, but C((n-k)+k-1, (n-k))/C(n+k-1, n) (C(n,k) being the binomial coefficient) does not yield the correct answer.

Answer

Birthday Coverage is basically a Coupon Collector's problem.

You have $n$ people who drew birthdays with repetition, and wish to find the probability that all $365$ different days were drawn among all $n$ people. ($n\geq 365$)

$$\mathsf P(T\leq n)= 365!\; \left\lbrace\begin{matrix}n\\365\end{matrix}\right\rbrace\; 365^{-n}$$

Where, the braces indicate a Stirling number of the second kind.   Also represented as $\mathrm S(n, 365)$.