# choosing poker hand with a specific card

How many ways can you choose at least one A from a deck of card in a poker hand?

I just wanted to double check my answer, would it be

C(52,5)- C(48,5)

Help is much appreciated,

• Yes, your calculation gives the number of hands with at least one Ace. – André Nicolas Apr 23 '14 at 19:32

To get at least one ace is to get 1, 2, 3, or 4. You are selecting the aces among the four aces, the other cards among the $52 - 4 = 48$ non-aces. In all: $$\binom{4}{1} \cdot \binom{48}{4} + \binom{4}{2} \cdot \binom{48}{3} + \binom{4}{3} \cdot \binom{48}{2} + \binom{4}{4} \cdot \binom{48}{1}$$ Or you could say there are $\binom{52}{5}$ hands in all, of those $\binom{48}{5}$ are ace-less, which gives: $$\binom{52}{5} - \binom{48}{5}$$