Suppose we have $n$ bins and $k<n$ is some natural number. Each turn we select at random $k$ distinct bins out of the $n$ bins in which we place a ball. How many turns does it take on average until there is at least $1$ ball in each of the bins? This part has been solved in the comments (as it turned out to be a duplicate).
Related to this : If we call a collection of $k$ balls a winner if one of the balls is the first one to enter a bin, how many winners are there on average. So the first batch of ball has a 100$\%$ chance of being a winner and then the probability of being a winner decreases as more balls are added. For the coupon collector's problem mentioned in the comments (with $k=1$) it is obvious that there will always be exactly $n$ winning batches.