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?

14080 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-12-23T02:24:44

1287

This is a combinations question.

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

$^{n} C_{r} = \frac{n!}{r! \times (n - r)!}$ $""^nC_r = (n!)/(r!xx(n-r)!)$

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

$^{13} C_{5} = \frac{13!}{5! \times (13 - 5)!} = \frac{13!}{5! \times 8!}$ $""^13C_5 = (13!)/(5!xx(13-5)!)=(13!)/(5!xx8!)$

$= \frac{13 \times 12 \times 11 \times 10 \times 9}{5 \times 4 \times 3 \times 2 \times 1}$ $= (13xx12xx11xx10xx9)/(5xx4xx3xx2xx1)$

$= 13 \times 3 \times 11 \times 3$ $= 13xx3xx11xx3$

$= 1287$ $=1287$

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".