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Math Help - Factorial Partial Sum

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    Factorial Partial Sum

    I am unsure of how Wolfram Alpha computes the following partial sum:

    sum of (n-k)!/(n-r-k)! from 1 to n-r - Wolfram|Alpha

    If someone could explain to me how the equation for the partial sum is derived I'd greatly appreciate it. Thanks!
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  2. #2
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    Quote Originally Posted by cyclic1001 View Post
    I am unsure of how Wolfram Alpha computes the following partial sum:

    sum of (n-k)!/(n-r-k)! from 1 to n-r - Wolfram|Alpha

    If someone could explain to me how the equation for the partial sum is derived I'd greatly appreciate it. Thanks!
    First, simplify...

    n - k - r < n - k, so


    (n - k)! = (n - k)(n - k - 1)(n - k - 2)\dots (n - k - r) \dots 3 \cdot 2 \cdot 1

     = (n - k)(n - k - 1)(n - k - 2)\dots (n - k - r)!.


    Therefore

    \frac{(n - k)!}{(n - k - r)!} = \frac{(n - k)(n - k - 1)(n - k - 2)\dots (n - k - r)!}{(n - k - r)!}

     = (n - k)(n - k - 1)(n - k - 2) \dots (n - k - r + 1).



    Can you now try to work out

    \sum_{k = 1}^{n - r} (n - k)(n - k - 1)(n - k - 2) \dots (n - k - r + 1)?
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