Please give me some hints about this problem.
Express the sum
$S = \binom{99}{33} + \binom{100}{34} + \ldots + \binom{200}{134}$
in this form
$S = \binom{}{} - \binom{}{}$
I try using the following relation
$\binom{r}{k} = \frac{r}{k}\binom{r-1}{k-1}$
then
$\binom{r}{k} = \binom{r-1}{k-1} - \frac{k-r}{k}\binom{r-1}{k-1}$
Am I on the right track? Any suggestions would be appreciated!
Recall the hockey stick identity \begin{eqnarray*} \sum_{i=r}^{n} \binom{i}{r}=\binom{n+1}{r+1}. \end{eqnarray*} Your sum can be written as \begin{eqnarray*} \sum_{i=99}^{200} \binom{i}{66}=\sum_{i=66}^{200} \binom{i}{66}-\sum_{i=66}^{98} \binom{i}{66}=\color{red}{\binom{201}{67}-\binom{99}{67}}. \end{eqnarray*}