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

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