I have 100 balls that are numbered and 100 boxes that are numbered. Each box can only contain one ball. The balls are randomly placed in the boxes. What is the probability that at least one ball ends up in the same numbered box?
I know that the number of ways I can choose the 100 balls is 100!
I'm not really sure where to go next. Would it be easier to look at the event that none of the balls are in the same numbered box?