The coefficient of $t^{20}$ in the expansion of $(1 + t^2)^{10}(1 + t^{10})(1 + t^{20})$ is

The coefficient of $t^{24}$ in the expansion of $(1 + t^2)^{12}(1 + t^{12})(1 + t^{24})$ is

If the $6^{th}$ term in the expansion of the binomial $[\sqrt{2^{\log(10 - 3^x)}} + \sqrt[5]{2^{(x - 2)\log 3}}]^m$ is equal to $21$ and it is known that the binomial coefficients of the $2^{nd}$,$3^{rd}$ and $4^{th}$ terms in the expansion represent respectively the first,third and fifth terms of an $A.P.$ (the symbol $\log$ stands for logarithm to the base $10$),then $x = $

If $(1+x+x^2)^n = c_0 + c_1 x + c_2 x^2 + \ldots$,then the value of $c_0 c_1 - c_1 c_2 + c_2 c_3 - \ldots$ is

Let the sixth term in the binomial expansion of $\left(\sqrt{2^{\log_2(10-3^x)}} + \sqrt[5]{2^{(x-2)\log_2 3}}\right)^m$,in the increasing powers of $2^{(x-2)\log_2 3}$,be $21$. If the binomial coefficients of the second,third,and fourth terms in the expansion are respectively the first,third,and fifth terms of an $A.P.$,then the sum of the squares of all possible values of $x$ is $.........$.

If the coefficient of $x^{30}$ in the expansion of $\left(1+\frac{1}{x}\right)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 ; x \neq 0$ is $\alpha$,then $|\alpha|$ equals