Question

a full house is a five-card hand with a three-of-a-kind and pair of different ranks. we say that a full house is split if the ranks are not consecutive. thus {3, 5} are split but {7, 8} are not. we say that a and k are consecutive. how many split full houses can be formed?

Answer 1

To form a split full house, we need a three-of-a-kind and a pair of different ranks. There are 13 ranks in a standard deck of cards. By calculating the number of choices for each step, we find that there are 3,744 split full houses that can be formed.

Step-by-step explanation:

To form a split full house, we need a three-of-a-kind and a pair of different ranks. There are 13 ranks in a standard deck of cards, ranging from 2 to 10, and the face cards J, Q, K, and A. Let's calculate the number of split full houses:

Step 1: Choose the rank for the three-of-a-kind. There are 13 choices for this rank.

Step 2: Choose the rank for the pair. Since we want the pair to be of a different rank from the three-of-a-kind, there are 12 remaining choices for this rank.

Step 3: Choose the suits for the three-of-a-kind and pair. For the three-of-a-kind, we have 4 choices for each of the 3 cards, so there are (4 choose 3) = 4 possible combinations. For the pair, we have 4 choices for each of the 2 cards, so there are (4 choose 2) = 6 possible combinations.

Step 4: Multiply the number of choices in each step to get the total number of split full houses: 13 * 12 * 4 * 6 = 3,744.

Therefore, there are 3,744 split full houses that can be formed.