Adam S. answered • 09/06/16

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There are 13 hearts in the deck, therefore 13 possible hearts for the first card. For each heart card, there are 5 clubs chosen without replacement. How many ways can 5 clubs be chosen from 13 total clubs without replacement when the order of the 5 clubs is not important.

number of hearts = 13

*

combination of clubs = (13 choose 5) = 13!/(13 - 5)!5!

-> from combinatronics formula: (n choose r) = n!/(n-r)!r!, where n! = n * (n -1) * (n - 2) *... 1.

combination of clubs = (13 * 12 * 11 * 10 *...)/(8 * 7 * 6 * ...)(5 * 4 * 3 *..)

=(13 * 12* 11 * 10 * 9)/(5 * 4 * 3 * 2 * 1) = 1287.

total number of combinations = 13 * 1287 = 16731 ways.