# Playing Cards – Four of a Kind

**Problem**

5 cards are randomly drawn from a standard 52 card deck. What is the probability to get four of a kind. For example 4 different aces plus some additional card

**Solution**

The total number of ways to choose 5 cards from 52 cards is straight forward and given by the formula n choose k where n = 52 and k = 5

(52 C 5) = (52!/(47! x 5!)) = 2598960

The tricky part in this question is calculating the total number of combinations that yield to four of a kind. Here is how we do that

**(1)** There are 13 ways to choose 1 number from 13 numbers (easy)

**(2)** Once a number is chosen there is no need to choose a color simply because there are 4 colors available and we need 4 but if you want to apply the formula then it is choosing 4 from 4 and there is only one way to do that. **(3)** The third step is to choose one number for the fifth card from the remaining 12 numbers **(4)** The last step is to choose a color for the fifth number from 4 colors available. So the total number of combinations is

(13 C 1) x (4 C 4) x (12 C 1) x (4 C 1) = 13 x 1 x 12 x 4 = 624

To calculate probability just divide the two numbers above

probability = 624/2598960