Combination of cards 
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 $1098240$.

We will use the notation $\binom{n}{r}$, which is more common among mathematicians,  where you write ${}_nC_r$.  
The kind of card that we have a pair of can be chosen in $\binom{13}{1}$ ways. For each choice of kind, the actual cards can be chosen in $\binom{4}{2}$ ways.
(By kind we mean things like Ace, or $7$, or Queen.)
For each choice made so far, we now count the number of ways to pick the rest of the cards. The kinds of cards we have singletons of can be chosen in $\binom{12}{3}$ ways. For each such choice, the actual cards can be chosen in $4^3$ ways. This is because if, for example, we are to have a $7$, a $10$, and a Queen, the $7$ can be picked in $4$ ways, as can the $10$, as can the Queen.   The total number of one pair hands is therefore 
$$(13)\binom{4}{2}\binom{12}{3}4^3.$$
Compute. We get $1098240$.
