# Combinatorial Identity

1. Mar 7, 2013

### icystrike

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:

$$\sum^{r-1}_{k=0} (^{n-1}_{r-(k+1)}) \times (^{r}_{k}) = (^{n+r-1}_{r-1})$$

Much thanks :)

2. Mar 8, 2013

### CompuChip

Last edited by a moderator: May 6, 2017
3. Mar 8, 2013

### icystrike

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 $(x_{1},x_{2},...,x_{r})$ satisfying:

$x_{1}+x_{2}+...x_{r}=n$​

Last edited by a moderator: May 6, 2017