How many hands are there with exactly 5 hearts after drawing 7 cards from a deck?

Question

Draw 7 cards from a deck of 52 cards. How many hands are there with exactly 5 hearts?

Will it be something like $$\frac{1!}{(52!51!50!49!48!)\cdot(7!6!5!4!3!)}$$ I'm pretty sure its wrong, any help will be appreciated!

Answer 1

There are $13$ hearts. You have to choose $5$ from them and the other $2$ from the rest of the deck which comprises of $39$ cards. Hence the total number of hands of $7$ cards with $5$ hearts will be $$\binom{13}{5}\binom{39}{2}$$ The total number of hands possible is $\binom{52}{7}$. Hnece if you want the probability that a hand of $7$ cards had $5$ hearts it will be $$\frac{\binom{13}{5}\binom{39}{2}}{\binom{52}{7}}$$

Answer 2

For a hand with fixed non-heart cards, we have ${13 \choose 5}$ possible hands where only the hearts vary. In turn, there are $\frac{39 \cdot 38}{2}$ ways of fixing the non-heart cards before we vary the heart cards.

So the answer is $\frac{39 \cdot 38}{2} \cdot {13 \choose 5}$ (which is the same as the other answer)