Prove an equality using combinatorial arguments $$n \cdot {2^{n - 1}} = \sum\limits_{k = 1}^n {k\left( {\begin{array}{*{20}{c}}
   n  \\
   k  \\
\end{array}} \right)} $$
The left-hand side can describe the number of possibilities choosing a committee with one chairman. 
How can the right-hand side feet to this story?

It is equivalent because for each group with the size of $k$ we can choose $k$ different chairmans. 
