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

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}} = $

If $\left({ }^{30} C _1\right)^2+2\left({ }^{30} C _2\right)^2+3\left({ }^{30} C _3\right)^2+\ldots \ldots+30\left({ }^{30} C _{30}\right)^2=$ $\frac{\alpha 60 !}{(30 !)^2}$, then $\alpha$ is equal to

The coefficient of $x^9$ in the polynomial given by $\sum\limits_{r – 1}^{11} {(x + r)\,(x + r + 1)\,(x + r + 2)…\,(x + r + 9)}$ is

If ${}^{21}{C_1} + 3.{}^{21}{C_3} + 5.{}^{21}{C_5} + ……19{}^{21}{C_{19}} + 21.{}^{21}{C_{21}} = k$ Then number of prime factors of $k$ is