Let's say we have a playlist consisting of $N$ songs. Each time I want to listen to music, I pick a random sample of $k\leq N$ different songs ($k$ is constant and deterministic over time) from that playlist and listen to them. Note that in the $k$ songs of one sample there are no duplicates, but the next may contain songs I've already heard in other samples. What is the expected number of samples I have to pick such that I have heard each song of that playlist at least once?
I calculated a simple example, where $N=3, k=2$ and let $X$ denote the random variable representing the number of samples it takes to cover all songs. Then $$ E[X]=\sum_{n=1}^{\infty}n\cdot Pr(X=n)=0+2\cdot\frac{2}{3}+3\cdot\frac{1}{3}\cdot\frac{2}{3}\dots=\sum_{n\geq 2} n\cdot\frac{2}{3}\cdot\frac{1}{3^{n-2}} \\ =\frac{2}{3}\cdot\sum_{n\geq 0}\frac{n+2}{3^n}=\frac{2}{3}\cdot\left(\frac{3}{(3-1)^2}+\frac{2}{3-1}\right)= \frac{7}{6} $$
But if I try to be more general or even only consider a little more difficult example I have problems calculating the individual probabilities as the possible combinations for $X=n$ depend on the previous combinations and I don't know how to take that into account.
Can anybody help me out with that and maybe also verify if my calculations for the example were correct?
