Summation of combinations 
Possible Duplicate:
simple binomial theorem proof 

Why is $${6\choose 0} + {7\choose 1} + \ldots + {n+6 \choose n} = {n+7 \choose n}\;?$$

The standard recursive formula for binomial coefficients is
$\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$
There is a technique for recursion called "unrolling," where you repeatedly substitute a recurrence into its equation for itself. Here if we reverse the terms and then substitute repeatedly we get:
$\binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}$
$\binom{n}{k} = \binom{n-1}{k} + \binom{n-2}{k-1} + \binom{n-2}{k-2}$
$\vdots$
$\binom{n}{k} = \binom{n-1}{k} + \binom{n-2}{k-1} + \binom{n-3}{k-2} + \cdots + \binom{n-k}{0}$
