If $${({}^{30}{C_1})^2} + 2{({}^{30}{C_2})^2} + 3{({}^{30}{C_3})^2}\, + \,...\, + \,30{({}^{30}{C_{30}})^2} = {{\alpha 60!} \over {{{(30!)}^2}}}$$ then $$\alpha$$ is equal to :

The value of $$\sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_r}} $$ is

$$\sum\limits_{r=1}^{20}\left(r^{2}+1\right)(r !)$$ is equal to

The remainder when $$7^{2022}+3^{2022}$$ is divided by 5 is :