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Deduce this combinatorial summation formula watch

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    I am working on a combinatorial summation problem (please see the attached file).

    In my Combinatorics lectures, we studied a theorem relating to orthogonality for binomial coefficients. This is what the first sentence of the problem relates to.

    I have managed to solve the first part of the question (finding the Stirling number). However, I am unsure how to solve the very last part of the problem (which requires a deduction).
    I would greatly appreciate a HINT(S) as to how to solve this last part of the problem.
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    (Original post by CKKOY)
    I am working on a combinatorial summation problem (please see the attached file).

    In my Combinatorics lectures, we studied a theorem relating to orthogonality for binomial coefficients. This is what the first sentence of the problem relates to.

    I have managed to solve the first part of the question (finding the Stirling number). However, I am unsure how to solve the very last part of the problem (which requires a deduction).
    I would greatly appreciate a HINT(S) as to how to solve this last part of the problem.
    Not really my area, but wikipedia suggests that there is a formula

    { m\brace k} = \frac{1}{k!} \sum_{j=0}^{k} {k \choose j} j^{m}

    that looks like you can punch around to get the result that you require. Did this arise in the bits of the question you've done already?
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    Thank you for the reply. Yes, it did come up in the other parts of the question.
 
 
 
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