The total # of possible outcomes is 52 choose 7 = 52!/(45!*7!) = 133784560. The number of ways to get exactly 3 aces is (4 choose 3) * (49 choose 4) = 4!/(3!1!) * 49!/45!4! = 847504. Note that the "4 choose 3" is the number of combinations of 3 aces out of 4 aces in the deck, and the "49 choose 4" is the number of combinations of 4 additional cards out of the remaining 49 cards left in the deck after the 3 aces.

So the probability you seek is 847504/133784560 = .006334842. This is different than 9/1547 = .005817712

How did you arrive at 9/1547?