Probability of getting all four aces playing with 13 people

Question

There are thirteen people sitting on a table. The dealer deals them a card each from his full deck(52 cards). He repeats this until all of his cards have been used. What is the probability that one of the players gets all four of the aces?

Answer 1

One player will have the ace of hearts and three other cards.

There is then a $\tfrac{3}{51}$ chance that one of those three cards will be an ace of diamonds.

Given that, there is a $\tfrac{2}{50}$ conditional chance that one of that players two remaining cards is the ace of spades.

Finally, there is a $\tfrac{1}{49}$ conditional chance that the remaining card turns out to be the ace of clubs.

So the chance that all three aces are in the same hand is: $\tfrac{3\cdot2\cdot1}{51\cdot50\cdot49}$.

$$ \text{Answer} = \frac{1}{20825} $$