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Math Help - Binomial Theorem Again

  1. #1
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    Binomial Theorem Again

    Hello,

    I've been asked to derive a closed expressions for the following sum:

    \sum_{i=m}^n (-1)^i {n\choose i}{i\choose m}

    (n= 0,1,2,.....) and (m=1,2,3,...,n)

    if been given the following hints but dont understand how to use the third one. Any help would be appreciated.

    1. Prove that is equal to zero if and is equal to 1 if .

    2. Prove that for .

    3. Deduce from 1. and 2. that your expression is equal to zero if and is equal to 1 if .
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  2. #2
    Moo
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    Hello,


    From 2., you can rewrite  {n \choose i} {i \choose m}

    Then, there will be a factor that won't depend on i. So you will be able to get it out of the sum !

    And thereafter, try to use 1.
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