How many five card hands (drawn from a standard 52-card deck) containing a two-of-a-kind?

Answer 1

Consider any of the card rank say aces there are 4 aces and ₂C₂ =6 ways to arrange 2 aces and there are 13 groups 2's 3'.....Q's and K's. So the number of ways to get 2 of a kind is 6*13=78.

There are 50 cards left after we get our pair so there are ₅₀C₃=19,600 ways to arrange the remaining three cards.

The number of 5 card hands containing 2 off a kind is 78*19,600=1,528,800 hands.

Jim