After dividing a standard deck of cards into 4 equal sized piles, what's the probability that exactly one ace is in each pile? I've had a couple of ideas about how to set this problem up but nothing seems to come out correctly. For instance, I can look at the probabilities that each pile has a single ace and set this up as a multiplication of conditional probabilities. It also occurred to me that I could view the events as (the event that Ace of Spades and Ace of Hearts are in different piles), (the event that Ace Spades, Diamonds, and Hearts are in different piles), and (the event that all aces are in different piles). However, I'm having a hard time even determining what the first of these probabilities should be.
Is it correct that the probability of the first pile having a single ace is $\dfrac{\binom{4}{1}\binom{48}{12}}{\binom{52}{13}}$? It's been a while since I've done probability so I'm having a little trouble getting started, though I know that eventually I'm going to multiply a number of conditional probabilities together.