What will be the sum of the series of binomial co-efficients? What will be the sum of the following binomial co-efficent series 
$$\binom{z+1}{z} + \binom{z+2}{z} + \binom{z+3}{z} + \dots + \binom{z+r}{z} = \sum\limits_{i=1}^r \binom{z+i}{z}$$
Thank you

Hint Add $\binom{z+1}{z+1}$ to you sum and use $\binom{k}{z}+\binom{k}{z+1}=\binom{k+1}{z+1}$ repeatedly:
$$\binom{z+1}{z+1}+\binom{z+1}{z} + \binom{z+2}{z} + \binom{z+3}{z} + \dots + \binom{z+r}{z} \\
=\binom{z+2}{z+1}+\binom{z+2}{z} + \binom{z+3}{z} + \dots + \binom{z+r}{z} \\=...$$
Here I am using the standard notation $\binom{k}{z}$ for $_kC_z$.
P.S. You can also see this formula backwards: using
$$\binom{z+k}{z}=\binom{z+k+1}{z+1}-\binom{z+k}{z+1}$$
you sum becomes telescopic.
