OK, so here's a simple problem: Suppose I roll a D6 one hundred times. What is the probability of any number appearing exactly 50 times?
Now, the probability of rolling 50 ones is presumably given by the number of 100-roll sequences that contain exactly 50 ones divided by the total number of possible sequences. But I could roll 50 twos instead. Which suggests that you need to add up all six ways to meet the criteria... except that these cases are not mutually-exclusive. It's possible, for example, to roll 50 threes and 50 sixes. Freakishly unlikely, but possible.
So... um... how do I attack this?