How many 5 card hands contain exactly 2 queens

There are 4 queens in the deck, hence (4 choose 2) = 4! / (2! (4-2) !) = 6 ways of getting 2 queens.

The other 3 cards in the deck can be any of the remaining 48 non-queens; there are (48 choose 3) = 48! / (3! (48-3) !) = 17,296 ways of getting these.

In total, there are (4 choose 2) * (48 choose 3) = 103,776 total possible hands.