Suppose we have m objects and we draw one uniformly n times with replacement. Some objects will be drawn at least once, some never. What is the expected value of the number of objects that are drawn, for certain m and n?
Phrased differently: We have m days in a year. We have n people. There are some days in a year when somebody has birthday, and there are ordinary days. What is the expected value of the number of days in the year when someone has birthday? The number of days in a year is also a parameter.
Once we have solved that, another problem. We want to have as many collisions as possible. What is the probability that the birthday set is smaller? Suppose that you are a teacher and you don't like birthdays. The ideal scenario would be if all the children in the class had birthday on the same day of the year. If that's not possible, then we want at least as many kids share birthdays as possible. You are a teacher and you have k classes to choose (randomly). What is the best (smallest) birthday set achievable for m days, n kids and k tries? I am asking for the expected value.