Ways to Pick a Six-Card HandDate: 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/ |
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