I suck at probability, so I thought I would ask here.

To clarify, given a population of size P, a sample size of K, and an arbitrary number of trials N, how do I compute the probability of having included each member of the population at least once in the experiment?

This problem is difficult to wrap my head around. It seems like it uses a combination of combinatorics and dependent events, which really throws me off.

Edit: This problem isn't the coupon collector's problem (please see some of my responses below). Think of the coupon collector's problem as being a special case of this problem where K = 1. My question is meant to cover an arbitrary K >= 1.