You are dealt 6 cards. How many ways can you be dealt 1 heart and 5 clubs?

Question

You are dealt 6 cards. How many ways can you be dealt 1 heart and 5 clubs?

Answer 1

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.