Laura M. answered • 05/09/19

Approachable, Experienced Math, Test Prep, and Chemistry Tutor

First, find the number of possible 5 card hands. This is 52 choose 5, or 52! (read 52 factorial) divided by [47!* 5!] (the * here for clarity) 5!=5*4*3*2*1, so this is 52*51*50*49*48 / 120

(You can also think of this as picking cards without replacement one at a time from a deck, and dividing by the number of ways those cards can be arranged). This number is the denominator for the given problem.

Next pick the card value for the pairs. 13 options for the first choice, 12 for the second. For each pair, you have 4 choose 2 = 4*3/2=6 options for the suits. Multiply these numbers together: 13*12*6*6. Lastly, the fifth card needs to be chosen. It cannot match either of the pairs, so there are 52-8=44 cards to pick from. The numerator is then 13*12*6*6*44.