Deck of cards combinations question?

A standard deck consists of 52 cards. How many 5-card hands are possible if the cards are all hearts?

Really confused how to do these type of problems. Would it be 5148?

1 Answer
Dec 23, 2017

1287

Explanation:

This is a combinations question.

The number of ways you can combine n objects taken r at a time is given by:

""^nC_r = (n!)/(r!xx(n-r)!)

Although there are 52 cards in a deck, only 52/4 =13 of them are hearts. So in this example n=13 and r=5.

""^13C_5 = (13!)/(5!xx(13-5)!)=(13!)/(5!xx8!)

= (13xx12xx11xx10xx9)/(5xx4xx3xx2xx1)

= 13xx3xx11xx3

=1287

So, the number of hands possible consisting of only hearts is 1287.

NB: In a hand of cards the order in which the cards are dealt is not relevant, hence a single hand is one "combination".