From a deck of 52 cards, how many five card poker hands can be formed if there is a pair (two of the cards are the same number, and none of the other cards are the same number)?
I believe you can pick out the first card by ${_4}C_2$, as there are 4 cards which would be the same number (as there are 4 suits). I would pick two from here.
From here on though, I am unsure. I believe it involves the numbers 48, 44, and 40, as after every picking you cannot have an identical number anywhere, so there would be 4 less to choose from. However, I don't believe I can just do $_{48}C_1 * _{44}C_1,..$ as I am not simply selecting 1 card from 48 and removing 4 random ones.
The answer is $1098240$.