Counting the number of ways of getting 4 aces

Question

Suppose I am playing bridge with 3 others, and I want to find the number of ways I could get $4$ aces. The 13 cards are dealt to me first.

I had two solutions I thought of, but the numbers are different.

In the first one, well, I have 13 spots for cards, and I choose 4 of the spots in which aces are placed. Hence, $\binom {13} 4$.

In the second solution, I thought; for me, there are $\binom {52} {13}$ ways in which I could get cards, and for each of those ways, there are $\binom {13} 4$ ways in which I could get 4 aces, so the answer is $\binom {52} {13} \cdot \binom {13} 4$.

Which answer is correct?

Answer 1

Since order doesn't matter, the spots in which the aces are placed in your hand don't matter. Your hand is completely characterized by the $9$ non-aces it contains, and those cards can be chosen (in any order) from any of the $48$ non-aces in the deck. Therefore, there are $\binom{48}{9}$ possible hands that contain all $4$ aces.

Your first solution is wrong because (a) it makes order matter (by determining where in the hand the aces are placed), when you've said it doesn't, and (b) it fails to distinguish among the various combinations of non-aces that your hand could contain.

Your second solution also is wrong because it multiplies the number of all possible hands (a number that is already bigger than the number of all hands with $4$ aces) by the number of spots in the hand that you can place the aces.

Answer 2

Hint

As has been remarked, "number of ways" can be interpreted in many ways.

Taking it to mean the number of ways you would use for computing the probability of getting such a hand,(which btw is a well defined term in bridge) think of the hypergeometric distribution.

Further hint

P(you get a hand with $k$ aces and $(13-k)$ non-aces = $\dfrac{\binom4{k}\binom {48}{13-k}}{\binom{52}{13}}$

FInal Hint

The "number of ways" you seek for this type of hand is given by the numerator of the expression, you should easily be able to work out the number of ways for the particular type of hand you seek.