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May 20th, 2011, 05:17 AM   #1
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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^n-r, of the term x^r*y^n-r.
Now those r factors can be chosen in nCr ways. So the co-efficient of x^r*y^n-r is nCr.

I know this is right but I don't understand how...

How come the co-efficient is coming as nCr?

I don't understand how the co-efficient and the combination of r among n factors are linked?

Cheers
R
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May 20th, 2011, 08:38 AM   #2
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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 .
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May 20th, 2011, 04:22 PM   #3
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Re: Binomial expansion as combination

Thanks a lot, I got it

Cheers
R
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