1.  Hands with no identical cards

C(52,5) = 2598960

2. Hands with exactly 1 pair of identical cards

Choose the pair of identical cards 52 ways. Choose the other 4 cards
in C(51,3)

That's 52*C(51,3) = 1082900

3. Hands with two exactly two identical cards.

Choose the two types of identical cards C(52,2) ways

Choose the 5th car 50  ways.

That's C(52,2)*50 = 66300 ways.

2598960 + 1082900 + 66300 = 3748160 hands 

Edwin