Question

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?

Answer 1

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".