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#1

$Old$ August 27th, 2008, 06:26 AM

Number Cruncher 20 $Number Cruncher 20 is offline$

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$Default$ Binomial Theorem Problem

Hello,

I am having trouble deriving a simple closed form expression for the following sum:

Sn =

for n = 0,1,2,3,4,.....

I am not sure how to use the binomial theorem to derive the expression. Any help would be much appreciated.

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#2

$Old$ August 27th, 2008, 06:30 AM

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Hello,

Your sum can be written $S_n=\sum_{i=0}^n\binom{n}{i}(-1)^i1^{n-i}$ which looks like $\sum_{i=0}^n\binom{n}{i}x^iy^{n-i}=(x+y)^n$ , doesn't it ?

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$Old$ August 27th, 2008, 06:48 AM

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$Default$ Binomial Theorem Problem

Yes I see it now, thanks. By any chance would you know how to get started on proving that:

for

.

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$Old$ August 27th, 2008, 07:01 AM

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Originally Posted by Number Cruncher 20 $View Post$

By any chance would you know how to get started on proving that:

for

.

This is straightforward. We have

$\binom{n}{k}\binom{k}{l}=\frac{n!}{k!(n-k)!}\cdot \frac{k!}{l!(k-l)!}$ and $\binom{n}{l}\binom{n-l}{k-l}=\frac{n!}{l!(n-l)!}\cdot \frac{(n-l)!}{(k-l)!(n-l-(k-l))!}$

If you simplify these two expressions you'll see that they are equal.

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