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### Ways to Pick a Six-Card Hand

```
Date: 02/26/2001 at 06:26:06
From: Stephen Shaw
Subject: Combinations

Sir,

You are picking six cards from a deck, where suit is ignored (i.e.
there are 13 possible cards Ace-King, but you can only have at most 4
of each card in your hand of 6 picked). How many possible hands are
there? We are ignoring order, i.e. 2,4,6,8,9 and 4,2,9,8,6 are counted
as just 1 hand.

I would be very grateful for the answer, as I can't find it out from
anyone.

Thanks,
Steve
```

```
Date: 02/26/2001 at 11:48:13
From: Doctor Anthony
Subject: Re: Combinations

You can use a generating function to answer this. You require the
coefficient of x^6 in the expansion of:

(1 + x + x^2 + x^3 + x^4)^13

(1 - x^5)^13
------------
(1-x)^13

= (1 - 13x^5 + ... )[1 +C(13,1)x + C(14,2)x^2 + C(15,3)x^3
+ C(16,4)x^4 +C (17,5)x^5 + C(18,6)x^6 + ... ]

and picking out the terms in x^6 we get:

C(18,6)x^6 - 13.C(13,1)x^6

and the coefficient is

C(18,6) - 13 x 13  =  18564 - 169

=  18395

So there are 18395 different hands of six cards not taking the suit
into account.

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

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