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Math Help - Binomial Theorem Problem

  1. #1
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    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. #2
    Super Member flyingsquirrel's Avatar
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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<br /> <br />
, doesn't it ?
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  3. #3
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    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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  4. #4
    Super Member flyingsquirrel's Avatar
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    Quote 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)!}<br />
and <br />
\binom{n}{l}\binom{n-l}{k-l}=\frac{n!}{l!(n-l)!}\cdot \frac{(n-l)!}{(k-l)!(n-l-(k-l))!}<br />

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