the following real life example of personal significance got me curious about what the answer would look like:
Hasbro is currently selling a series of 24 different "My Little Pony" figurines, all sold separately in non-transparent bags with no clue as to which of the 24 ponies are inside. I only want six specific ponies, and the problem is knowing how many bags I would have to buy in order to get, with probability p, at least one of *each* of the six desirable ponies. Assume that there is an infinite supply of bags, and that the pony inside each bag is random. Assume further that I open all bags at once, not going back to the store for more.
To generalize and put the problem in more familiar terms:
Consider a die with n sides, numbered from 1 to n. How many throws must one make in order to have the chance p of getting all numbers [1..k] (where k <= n)?