Please someone explain to me how to do the following problem:
How many different 5 card poker hands can you get where one of the cards must be the ace of spades and another the ace of hearts?
My answer is $$\binom{52}{4} + \binom{51}{3}$$ because there are total of $52$ cards with $4$ aces. There are $\binom{52}{4}$ ways to choose the ace of spade. After we choose the ace of spade, there are total of $51$ cards left in the deck with $3$ aces. There are $\binom{51}{3}$ ways to choose the ace of hearts. Then we add them both.
Is my answer right?
Best Answer
Once you have selected the ace of spades and the ace of hearts, you have 50 remaining cards from which you choose 3 to complete the 5-card hand. Thus the total number of ways is simply $$\binom{50}{3}.$$