In a poker hand of 5 cards how many ways can you get a full house? (3 of a kind and a pair)
To calculate the number of ways to get a full house in a 5-card poker hand (3 of a kind and a pair), you can follow these steps:
Step 1: Determine the number of ways to choose the rank of the 3 of a kind.
There are 13 ranks in a standard deck of cards (2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace). You need to select one of these ranks as the rank for the 3 of a kind. Therefore, there are 13 possibilities for this step.
Step 2: Determine the number of ways to choose 3 cards of the chosen rank.
For the rank you selected in Step 1, there are 4 cards of that rank in the deck (one in each suit - hearts, diamonds, clubs, and spades). So, you need to choose 3 cards from those 4. The number of ways to do this is given by the combination formula: C(4, 3) = 4.
Step 3: Determine the number of ways to choose the rank of the pair.
After selecting the rank for the 3 of a kind in Step 1, there are 12 remaining ranks. However, for the pair, you need to choose a different rank from the one you selected for the 3 of a kind. So, there are 12 possibilities for this step.
Step 4: Determine the number of ways to choose 2 cards of the chosen pair rank.
For the rank you selected in Step 3, there are 4 cards of that rank in the deck. However, you only need to choose 2 cards from those 4. So, the number of ways to do this is given by the combination formula: C(4, 2) = 6.
Step 5: Calculate the total number of ways to get a full house.
To get the total number of ways to get a full house, you need to multiply the results from Steps 1, 2, 3, and 4 together:
13 * 4 * 12 * 6 = 3,744.
Therefore, there are 3,744 ways to get a full house in a 5-card poker hand.