5 How many poker hands contain a full house without any diam

5.

How many poker hands contain a full house without any diamonds? (For instance, Jsp Jcl Jhe 7he 7cl is one such hand.)

Solution

Full House – a hand consisting of one pair and a three-of-a-kind (3OAK) of a different rank than the pair

Since a full house has the form of one pair plus a three-of-a-kind then there are 13 * 12 = 156 choices for the ranks of the pair and the 3OAK

Since there can not be any Diamonds thus there are ³C₂ = 3 choices for the pair in its rank and ³C₃ = 1 choice for the 3OAK. Therefore there are 12 * 13 * ³C₂ * ³C₃ = 468 possible full houses.without any diamonds.