Case 1. The hand contains 3 aces and 2 non-aces.
There are 4 aces in the deck so there are C(4,3) = 4 ways to pick the 3 aces.
There are 48 non-aces in the deck, so you can pick the two non-aces
C(48,2)=1128 ways.
That's C(4,3)C(48,2) = (4)(1128) = 4512 ways.
Case 2. The hand contains all 4 aces and 1 non-ace.
There are 4 aces in the deck so there is C(4,4) = 1 way to pick all 4 aces.
There are 48 non-aces in the deck, so you can pick the one non-ace C(48,1) = 48
ways.
That's C(4,4)C(48,1) = (1)(48) = 48 ways.
Add them together: 4512+48 = 4560 ways.
Edwin
