If $1^2 \cdot ^{20}C_1 + 2^2 \cdot ^{20}C_2 + 3^2 \cdot ^{20}C_3 + \dots + 20^2 \cdot ^{20}C_{20} = A(2^\beta)$,then the ordered pair $(A, \beta)$ is equal to

Let ${ }^{n} C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+ x )^{ n }.$ If $\sum_{ k =0}^{10}\left(2^{2}+3 k \right){ }^{10} C _{ k }=\alpha \cdot 3^{10}+\beta \cdot 2^{10},$ where $\alpha, \beta \in R,$ then $\alpha+\beta$ is equal to ....... .

If $(1+x)^n=C_0+C_1 x+C_2 x^2+\ldots+C_n x^n$,then $C_0+2 C_1+3 C_2+\ldots+(n+1) C_n$ is equal to

If $n \geq 2$ is a positive integer,then the sum of the series ${ }^{n+1} C_{2}+2\left({ }^{2} C_{2}+{ }^{3} C_{2}+{ }^{4} C_{2}+\ldots+{ }^{n} C_{2}\right)$ is ...... .

$\sum_{k=0}^{20} \left({}^{20}C_{k}\right)^{2}$ is equal to :

If ${C_0}, {C_1}, {C_2}, ......., {C_n}$ are the binomial coefficients,then $2.{C_1} + {2^3}.{C_3} + {2^5}.{C_5} + ....$ equals