${10 \choose 4}+{11 \choose 4}+{12 \choose 4}+\cdots+{20 \choose 4}$ can be simplified as which of the following? ${10 \choose 4}+{11 \choose 4}+{12 \choose 4}+\cdots+{20 \choose 4}$ can be simplified as ?
A. ${21 \choose 5}$
B. ${20 \choose 5}-{11 \choose 4}$
C. ${21 \choose 5}-{10 \choose 5}$
D. ${20 \choose 4}$
Please give me a hint. I'm unable to group the terms.
By brute force, I'm getting ${21 \choose 5}-{10 \choose 5}$

What is the  problem in this solution?
If $S=\binom{10}{4} + \binom{11}{4} + \cdots + \binom{20}{4}$ we have
\begin{eqnarray}
S&=& \left \{ \binom{4}{4} + \binom{5}{4} \cdots + \binom{20}{4}\right \} - \left \{\binom{4}{4} + \binom{5}{4} \cdots + \binom{9}{4}\right \} &=& \binom{21}{5} - \binom{10}{5}
\end{eqnarray}
