The story:

A sport card store manager has $r$ customers, that together wish to assemble a $n$-cards collection. Every day, a random customer arrives and buys his favorite card (that is, each customer is associated with a single card), even if it has purchased the same card before. How many days will past before the customers complete their collection?

Formally, let $n\le r$ be integer parameters, and let $h:[r]\to [n]$ be a random projection from $[r]$ to $[n]$ (i.e., it maps every element uniformly and independently). How many random samples (with replacement) $(x,h(x))$ to we need to get before we see all $n$ values possible for $h$?

Clearly, there is some chance that $h$ is not onto and thus the expectation of the required number is not bounded.

I'm interested in a bound of the form:

- After $T(r,n,\delta)$ samples, with probability at least $1-\delta$, we have seen all possible $n$ values.

For example, if $r=3,n=2$, we have a probability of $1/4$ that $h$ is not onto, and if it is, then after collecting $4$ cards, the chance of not seeing both values is $(1/3)^4+(2/3)^4\le 0.21$. This means that $T(3,2,0.46)=4$ is a correct upper bound.