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May 20th, 2011, 05:17 AM  #1 
Member Joined: May 2011 Posts: 51 Thanks: 0  Binomial expansion as combination
Hi all, I m new here In one book the author says to look at the binomial theorem as the algebraic codification of a combinatorial process. In (x+y)^n=(x+y)*(x+y)*(x+y)...n times Consider them as 'n' numbers of factors, consider 'r' as a positive integer between 0 and n. Now any r numbers of factors among those n factors will give us x^r and the rest will give us y^nr, of the term x^r*y^nr. Now those r factors can be chosen in nCr ways. So the coefficient of x^r*y^nr is nCr. I know this is right but I don't understand how... How come the coefficient is coming as nCr? I don't understand how the coefficient and the combination of r among n factors are linked? Cheers R 
May 20th, 2011, 08:38 AM  #2 
Senior Member Joined: Feb 2010 Posts: 711 Thanks: 147  Re: Binomial expansion as combination
Look for example at . Expanded it equals Now look at just the term and think about where the came from. It could have come from the first parenthesis ... that is . It could have come from the second parenthesis ... that is . It could have come from the third parenthesis ... that is . It could have come from the fourth parenthesis ... that is . So the coefficient 4 is really just counting the possible arrangements of x's and y's in . That is, the coefficient is . 
May 20th, 2011, 04:22 PM  #3 
Member Joined: May 2011 Posts: 51 Thanks: 0  Re: Binomial expansion as combination
Thanks a lot, I got it Cheers R 

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