P & C

It is stated that there should only be 1 ace in each combination. So take out the 4 aces from the deck and determine the number of combinations out of the remaining 48 cards taken 4 at a time then multiply the result by 4.

$4\times ({ _{ 48 }{ C }_{ 4 }) }=4\times \frac { 48! }{ 44!\times 4! } =\boxed { 778320 }$

There are 4 ways of choosing the ace. The number of combinations of the remaining four cards is the number of combinations of 48 cards chosen 4 at a time, which is 48!/(44! 4!). So the number of poker hands containing exactly one ace are 4 48!/(44!*4!) = 778,320