If $\sum_{r=1}^{10} r !\left( r ^{3}+6 r ^{2}+2 r +5\right)=\alpha(11 !),$ then the value of $\alpha$ is equal to ...... .

The sum of coefficients in the expansion of ${(1 + x + {x^2})^n}$ is

For $x\, \in \,R\,,\,x\, \ne \, - 1,$ if ${(1 + x)^{2016}} + x{(1 + x)^{2015}} + {x^2}{(1 + x)^{2014}} + ....{x^{2016}} = \sum\limits_{i = 0}^{2016} {{a_i\,}{\,x^i}} ,$ then $a_{17}$ is equal to

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

If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + .... + {C_n}{x^n}$, then the value of ${C_0} + 2{C_1} + 3{C_2} + .... + (n + 1){C_n}$ will be

If $A$ denotes the sum of all the coefficients in the expansion of $\left(1-3 x+10 x^2\right)^n$ and $B$ denotes the sum of all the coefficients in the expansion of $\left(1+x^2\right)^n$, then :