How can I compute $\sum\limits_{k=100}^{200} \binom{k}{100}$?

Question

How can I compute $\sum\limits_{k=100}^{200} \binom{k}{100}$?

Answer 1

\begin{align*}\sum_{k=100}^{200}\binom{k}{100}&=\binom{100}{100}+\sum_{k=101}^{200}\left\{\binom{k+1}{101}-\binom{k}{101}\right\}\\ &=\binom{100}{100}+\binom{201}{101}-\binom{101}{101}\\&=\binom{201}{101}\end{align*}

The first equality is due to Pascal's identity: $$\binom{n-1}{r-1}+\binom{n-1}{r}=\binom{n}{r} \text{ for } 1\leq r\leq n-1$$

Answer 2

This is the hockey stick identity. There are many proofs and other nice identities here: https://artofproblemsolving.com/wiki/index.php/Combinatorial_identity