Deck of cards combinations question?

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?

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Answer 1 · 2017-12-23T02:24:44

1287

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".