Sum of odd terms is $A$ and sum of even terms is $B$ in the expansion ${(x + a)^n},$ then

If ${(1 + x - 2{x^2})^6} = 1 + {a_1}x + {a_2}{x^2} + .... + {a_{12}}{x^{12}}$, then the expression ${a_2} + {a_4} + {a_6} + .... + {a_{12}}$ has the value

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

If $^{20}{C_1} + \left( {{2^2}} \right){\,^{20}}{C_3} + \left( {{3^2}} \right){\,^{20}}{C_3} + \left( {{2^2}} \right) + ..... + \left( {{{20}^2}} \right){\,^{20}}{C_{20}} = A\left( {{2^\beta }} \right)$, then the ordered pair $(A, \beta )$ is equal to

Let $\left(\frac{n}{k}\right)=\frac{n !}{k !(n-k) !}$. Then the sum $\frac{1}{2^{10}} \sum \limits_{ k =0}^{10}\left(\frac{10}{ k }\right) k ^2$, lies in the interval

The coefficient of $t^{50}$ in $(1 + t^2)^{25} (1 + t^{25}) (1 + t^{40}) (1 + t^{45}) (1 + t^{47})$ is