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 :

If the sum of the coefficients in the expansion of ${({\alpha ^2}{x^2} - 2\alpha {\rm{ }}x + 1)^{51}}$ vanishes, then the value of $\alpha $ is

If the sum of the coefficients of all even powers of $x$ in the product $\left(1+x+x^{2}+\ldots+x^{2 n}\right)\left(1-x+x^{2}-x^{3}+\ldots+x^{2 n}\right)$ is $61,$ then $\mathrm{n}$ is equal to

The coefficient of $x^r (0 \le r \le n - 1)$ in the expression :

$(x + 2)^{n-1} + (x + 2)^{n-2}. (x + 1) + (x + 2)^{n-3} . (x + 1)^2; + ...... + (x + 1)^{n-1}$ is :

The expression $x^3 - 3x^2 - 9x + c$ can be written in the form $(x - a)^2 (x - b)$ if the values of $c$ is

The value of $^{15}C_0^2{ - ^{15}}C_1^2{ + ^{15}}C_2^2 - ....{ - ^{15}}C_{15}^2$ is