# Binominal cooefficient wording of a question

• Aug 29th 2012, 10:10 AM
uperkurk
Binominal cooefficient wording of a question
How many different 5-card hands can be dealt from a standard 52-card deck?

I've highlighted the word different because I believe it's the key to the question, am I right in thinking the answer is

$\displaystyle 52*51*50*49*48 = 311,875,200$
• Aug 29th 2012, 10:24 AM
Plato
Re: Binominal cooefficient wording of a question
Quote:

Originally Posted by uperkurk
How many different 5-card hands can be dealt from a standard 52-card deck?

If the question means "how many ways can you be dealt a five card hand?", then the answer is $\displaystyle \binom{52}{5}=2598960$.

Or does it have another meaning?
• Aug 29th 2012, 10:31 AM
uperkurk
Re: Binominal cooefficient wording of a question
Well I don't know that is just how the question is written but

23456
23465
23654
26543
65432

would all be classes as a different hand? I am not sure.
• Aug 29th 2012, 10:49 AM
Plato
Re: Binominal cooefficient wording of a question
Quote:

Originally Posted by uperkurk
Well I don't know that is just how the question is written but
23456
23465
23654
26543
65432
would all be classes as a different hand? I am not sure.

No. That is incorrect. Order makes no difference. Consider only content.
$\displaystyle \boxed{A\spadesuit}~\boxed{2\clubsuit}~\boxed{6 \spadesuit}~\boxed{10 \heartsuit}~\boxed{8 \diamondsuit}$ is the same hand in any order.
• Aug 29th 2012, 10:54 AM
HallsofIvy
Re: Binominal cooefficient wording of a question
No, they wouldn't. Your answer, 52*51*50*49*48, which is the same as $\displaystyle \frac{52!}{47!}$ or $\displaystyle _{52}P_5$ which includes the same cards in different orders. To not count the same cards in different orders as different hands, you need to divide by 5!, the number of orders of the same 5 cards. That will give you $\displaystyle \frac{52*51*50*49*48}{5!}$ which is exactly $\displaystyle \frac{52!}{5! 47!}= \begin{pmatrix}52 \\ 5\end{pmatrix}$ as Plato said.
• Aug 29th 2012, 11:06 AM
uperkurk
Re: Binominal cooefficient wording of a question
ok thanks for explaining :)