Combinatorics involving a poker hand

Question

How many 5 card poker hands can be dealt from a standard deck of 52 cards with 3 hearts and 2 diamonds?

I have tried this but I am not sure if it is correct:

${13\choose3}{13\choose2} = 22308$

Answer 1

That's correct. The general principle is the hypergeometric distribution, and the probability for $0$ spades, $3$ hearts, $2$ diamonds, and $0$ clubs is $$\frac{\binom{13}{0}\binom{13}{3}\binom{13}{2}\binom{13}{0}}{\binom{52}{5}}$$

Answer 2

Your attempt is correct. To specify a $5$-card poker hand with three hearts and to diamonds, you need to choose the three hearts present in $\binom{13}3$ ways, and then choose the two hearts present in $\binom{13}2$ ways. Since the hearts can be chosen independently of the diamonds, the total number of ways is found by multiplying the number of ways to do each.