If ${a_r}$ is the coefficient of ${x^r}$, in the expansion of ${(1 + x + {x^2})^n}$, then ${a_1} - 2{a_2} + 3{a_3} - .... - 2n\,{a_{2n}} = $

Coefficient of $x^{19}$ in the polynomial $(x-1) (x-2^1) (x-2^2) .... (x-2^{19})$ is

$\left( {\begin{array}{*{20}{c}}n\\0\end{array}} \right) + 2\,\left( {\begin{array}{*{20}{c}}n\\1\end{array}} \right) + {2^2}\left( {\begin{array}{*{20}{c}}n\\2\end{array}} \right) + ..... + {2^n}\left( {\begin{array}{*{20}{c}}n\\n\end{array}} \right)$ is equal to

The coefficient of $x ^{301}$ in $(1+x)^{500}+x(1+x)^{499}+x^2(1+x)^{498}+\ldots . .+x^{500}$ is:

If $A$ denotes the sum of all the coefficients in the expansion of $\left(1-3 x+10 x^2\right)^n$ and $B$ denotes the sum of all the coefficients in the expansion of $\left(1+x^2\right)^n$, then :

$\frac{{{C_0}}}{1} + \frac{{{C_2}}}{3} + \frac{{{C_4}}}{5} + \frac{{{C_6}}}{7} + ....$=