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

The sum of last eight consecutive coefficients in the expansion of $(1+x)^{15}$ is

$\sum_{\substack{i, j=0 \\ i \neq j}}^{n}{ }^{n} C_{i}{ }^{n} C_{j}$ is equal to

If ${a_k} = \frac{1}{{k(k + 1)}},$ for $k = 1,\,2,\,3,\,4,.....,\,n$, then ${\left( {\sum\limits_{k = 1}^n {{a_k}} } \right)^2} = $

The coefficient of $x^{49}$ in the expansion of $(x - 1)$$\left( {x\, - \,\frac{1}{2}\,} \right)$$\left( {x\, - \,\frac{1}{{{2^2}}}\,} \right)$ .....$\left( {x\, - \,\frac{1}{{{2^{49}}}}\,} \right)$ is equal to

The value of$^n{C_1}\sum\limits_{r = 0}^1 {^1{C_r}} { + ^n}{C_2}\left( {\sum\limits_{r = 0}^2 {^2{C_r}} } \right){ + ^n}{C_3}\left( {\sum\limits_{r = 0}^3 {^3{C_r}} } \right) + ......{ + ^n}{C_n}\left( {\sum\limits_{r = 0}^n {^n{C_r}} } \right)$ is equal to