A deck of 52 cards contains four aces. If the cards
are shuffled and distributed in a random manner to four
players so that each player receives 13 cards, what is the
probability that all four aces will be received by the same
player?
Answer is $$\dfrac{4\binom{13}{4}}{\binom{52}{4}}.$$
Why is $\binom{13}{4}$ taken in numerator?
Best Answer
Before dealing out the cards, give each player 13 coins. Now there are 52 coins on the table, and we will deal cards on top of each coin. We will choose four of the coins to receive aces, and all ${52\choose 4}$ ways of choosing which four that is are equally likely.
There are four ways of choosing who the lucky player is to get all four aces, and ${13\choose 4}$ ways of choosing which of that player's coins will get the aces.