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Q.
The value of $\displaystyle\sum_{m=0}^{2013} \displaystyle\sum_{n=0}^m\begin{pmatrix}2013 \\ m\end{pmatrix}\begin{pmatrix}m \\ n\end{pmatrix}$ equals
Binomial Theorem
Solution:
We have $\displaystyle\sum_{ m =0}^{2013}\left({ }^{2013} C _{ m }\right)\left[{ }^{ m } C _0+{ }^{ m } C _1+\ldots \ldots .+{ }^{ m } C _{ m }\right]\left(\right.$ opening $\left.\displaystyle\sum_{ n =0}^{ m }\right)$
$=\displaystyle\sum_{ m =0}^{2013}{ }^{2013} C _{ m } \cdot 2^{ m }={ }^{2013} C _0+{ }^{2013} C _1 \cdot 2+\ldots \ldots . .+{ }^{2013} C _{2013} \cdot 2^{2013}=(1+2)^{2013}=3^{2013}$