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