The probability of being dealt a 2 pair 5 card hand... Given a standard playing card deck of 52 cards, what is the probability of being dealt a 2 pair 5 card hand consisting exactly of one pair of face cards and one pair of NOT face cards is?

There are $\binom{52}{5}$ hands, all equally likely. 
For the number of hands that fit your description (the favourables), the kind of face card we have two of can be chosen in $\binom{3}{1}$ ways. For each of these ways, the actual cards can be chosen in $\binom{4}{2}$ ways. For each of these ways, the kind of non-face card we have two of can be chosen in $\binom{10}{1}$ ways, and the actual cards in $\binom{4}{2}$ ways. Finally, the useless fifth card can be chosen in $\binom{44}{1}$ ways, since we must avoid the kinds we have two of. That gives a total of $\binom{3}{1}\binom{4}{2}\binom{10}{1}\binom{4}{2}\binom{44}{1}$. 
