Evaluate:

We know that:

$\sum_{r=1}^{n}\left(\begin{array}{l}n \\ r\end{array}\right)=2^{n}-\left(\begin{array}{l}n \\ 0\end{array}\right)$

$\Rightarrow \sum_{r=1}^{6}\left(\begin{array}{l}6 \\ r\end{array}\right)=2^{6}-\left(\begin{array}{l}6 \\ 0\end{array}\right)$

$\Rightarrow \sum_{r=1}^{6}\left(\begin{array}{l}6 \\ r\end{array}\right)=64-1$

$\Rightarrow \sum_{r=1}^{6}\left(\begin{array}{l}6 \\ r\end{array}\right)=63$

Ans: $\sum_{r=1}^{6}\left(\begin{array}{l}6 \\ r\end{array}\right)=63$