Summation of binomial coefficients [duplicate]
Is there a closed formula for:
$\sum_{i=1}^{N}{\binom{i+k}{i}}$
( k is a constant whole number )
Solution 1:
Yes:
$$\sum_{i=1}^N\binom{i+k}i=\sum_{i=1}^N\binom{i+k}k=\sum_{i=0}^N\binom{i+k}k-1=\binom{N+k+1}{k+1}-1\;.$$