Suppose there are $a$ distinct kinds of badminton cards, each equally likely to be received with any given purchase, what is the expected number of purchases in order to acquire $b$ complete sets, i.e., at least $b$ of each of the $a$ types of cards$?$
This problem looks complicated to me. So I first considered the base case where $b=1$ it's quite simple and I got the answer as $$a\left(1+\frac12+\frac13+\cdots+\frac1a\right)$$
But now the problem occurs. I can't figure out how to find the expected number of purchases for $b=2$ or $b=3$ as for me the case of general $b$ is too difficult.
Any help will be greatly appreciated.