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Combinatorial Identity

  1. Mar 7, 2013 #1
    1. The problem statement, all variables and given/known data
    Hello PF! This is not a homework problem but I am curious to know if the following combinatorial identity exist:

    [tex] \sum^{r-1}_{k=0} (^{n-1}_{r-(k+1)}) \times (^{r}_{k}) = (^{n+r-1}_{r-1}) [/tex]


    Much thanks :)
     
  2. jcsd
  3. Mar 8, 2013 #2

    CompuChip

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    Last edited by a moderator: May 6, 2017
  4. Mar 8, 2013 #3
    Thank you! It is indeed Vandermonde's identity with n, m and r substituted with n-1, r, and r-1 respectively.

    I wrote this above identity as inspired by finding the number of distinct nonnegative integer-valued vectors [itex] (x_{1},x_{2},...,x_{r})[/itex] satisfying:

    [itex]x_{1}+x_{2}+...x_{r}=n[/itex]​
     
    Last edited by a moderator: May 6, 2017
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