Compute telescopic sum of binomial coefficients Is there a nice or simple form for a sum of the following form? 
$$ 1 + \sum_{i=1}^k \binom{n-1+i}{i} - \binom{n-1+i}{i-1}$$
Motivation: Due to a computation in the formalism of Schubert calculus the above sum with $k = \lceil n/2 \rceil -1$  is equal to the number of lines intersecting $2n-4$ general subspaces $H_j\subseteq \mathbb{P}^n$ of dimension $n-2$.

Hint
The following method does not use telescopic series but it makes use of elementary binomial theorem along with some geometric progression to arrive at the answer.
$$\begin{aligned}S&=\sum_{i=1}^{k}{n-1+i\choose n-1}-\sum_{i=1}^{k}{n-1+i\choose n}\\&=\left(\text{coeff. of } x^{n-1} \text{ in } \sum_{i=1}^{k}(1+x)^{n-1+i}\right)-\left(\text{coeff. of } x^{n} \text{ in } \sum_{i=1}^{k}(1+x)^{n-1+i}\right)\end{aligned}$$
