I have a stack of $n$ distinct cards. I randomly draw $N$ cards from the stack with replacement. Assume that $N \ge n$.
What is the probability that I observe all cards at least once?
I got this far:
The probability that the multiplicities of the states in the sample are $m_1, \dots, m_n$ is:
$$\frac{1}{n^{N}}\frac{N!}{m_1! \dots m_n!}$$
Therefore, the probability we want is found by evaluating the sum:
$$\frac{1}{n^{N}}\sum_{\begin{subarray}{c} m_{1}+\dots+m_{n}=N\\ m_{k}\ge1 \end{subarray}}\frac{N!}{m_{1}!\dots m_{n}!}$$
However, I'm stuck since I'm not sure how to evaluate this. I tried a change of variables $r_k = m_k - 1$, but this messes the factorials in the denominator.
Update: I found this paper, which contains a solution:
Kullback, Solomon. 1937. “On Certain Distributions Derived From the Multinomial Distribution.” The Annals of Mathematical Statistics 8(3): 127–44. http://projecteuclid.org/euclid.aoms/1177732409.
