$\left( {\begin{array}{*{20}{c}}n\\0\end{array}} \right) + 2\,\left( {\begin{array}{*{20}{c}}n\\1\end{array}} \right) + {2^2}\left( {\begin{array}{*{20}{c}}n\\2\end{array}} \right) + ..... + {2^n}\left( {\begin{array}{*{20}{c}}n\\n\end{array}} \right)$ is equal to
${2^n}$
$0$
${3^n}$
None of these
$\sum\limits_{n = 0}^4 {{{\left( {1009 - 2n} \right)}^4}\left( \begin{gathered}
4 \hfill \\
n \hfill \\
\end{gathered} \right)} {\left( { - 1} \right)^n}$ is
The sum of the coefficients of even power of $x$ in the expansion of ${(1 + x + {x^2} + {x^3})^5}$ is
If ${ }^{20} \mathrm{C}_{\mathrm{r}}$ is the co-efficient of $\mathrm{x}^{\mathrm{r}}$ in the expansion of $(1+x)^{20}$, then the value of $\sum_{r=0}^{20} r^{2}\,\,{ }^{20} C_{r}$ is equal to :
If ${(1 + x)^{15}} = {C_0} + {C_1}x + {C_2}{x^2} + ...... + {C_{15}}{x^{15}},$ then ${C_2} + 2{C_3} + 3{C_4} + .... + 14{C_{15}} = $
$\mathrm{b}=1+\frac{{ }^1 \mathrm{C}_0+{ }^1 \mathrm{C}_1}{1 !}+\frac{{ }^2 \mathrm{C}_0+{ }^2 \mathrm{C}_1+{ }^2 \mathrm{C}_2}{2 !}+\frac{{ }^3 \mathrm{C}_0+{ }^3 \mathrm{C}_1+{ }^3 \mathrm{C}_2+{ }^3 \mathrm{C}_3}{3 !}+\ldots$
Let $\mathrm{a}=1+\frac{{ }^2 \mathrm{C}_2}{3 !}+\frac{{ }^3 \mathrm{C}_2}{4 !}+\frac{{ }^4 \mathrm{C}_2}{5 !}+\ldots$, Then $\frac{2 b}{a^2}$ is equal to.........................