If I choose 5 cards at random, how many ways are there to get a full house?
To calculate the number of ways to get a full house when choosing 5 cards at random, we first need to understand what a full house is.
A full house is a poker hand that consists of three cards of the same rank and two cards of another rank. For example, a hand with three Kings and two Queens would be a full house.
To calculate the number of ways to get a full house, we can break it down into two steps:
Step 1: Choose the three cards of the same rank
To select the three cards of the same rank, we can choose any rank from the 13 available ranks (Ace, 2, 3, ..., 10, Jack, Queen, King). Once we choose a rank, there are four cards of that rank in a deck, so we have to choose three of them. We can compute this by using the combination formula: 13 choose 1 * 4 choose 3.
Step 2: Choose the two cards of the other rank
After choosing the three cards of the same rank, we need to select two cards of a different rank for the remaining two spots in the full house. Since we already selected one rank for the three cards, there are 12 remaining ranks to choose from. For the chosen rank, we have four cards available, and we need to choose two of them. Again, using the combination formula: 12 choose 1 * 4 choose 2.
Once we have chosen the ranks and the cards, we multiply the results from both steps together to get the total number of ways to get a full house:
Total number of ways = (13 choose 1 * 4 choose 3) * (12 choose 1 * 4 choose 2)
Evaluating this expression, we get:
Total number of ways = 13 * 4 * 1 * 4 * 12 * 6 = 3,744
Therefore, there are 3,744 ways to get a full house when choosing 5 cards at random.