Question

Find the number of 5-card stud poker hands in which all the cards are either spades or hearts.

Answer 1

I'm thinking that since there are a total of 26 spade and heart card all together, the answer should be 26*25*24*23*22

Answer 2

@epsilondelta you are overcounting greatly... once you determine one card, all others must be of the same kind i.e. \(26\times12\times11\times10\times9\)

Answer 3

even then you're still overcounting since you're distinguishing permutations of the same hand... you must divide by \(5!\). Essentially, the number of hands in which all cards are of a particular suit is given \(_{13}C_5=\dfrac{13!}{8!5!}=\dfrac{13\times12\times11\times10\times9}{5\times4\times3\times2}=13\times11\times9\) and so the number of hands in which all cards are of either particular suit is just \(2\times\:_{13}C_5=2\times13\times11\times9\)