The ratio of the coefficient of $x^2$ to the coefficient of $x^{10}$ in the expansion of $(x^5 + 4 \cdot 3^{-\log_{\sqrt{3}}\sqrt{x^3}})^{10}$ is

If $n$ is the degree of the polynomial,$\left[ {\frac{1}{{\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }}} \right]^8 + \left[ {\frac{1}{{\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} }}} \right]^8$ and $m$ is the coefficient of $x^{12}$ in it,then the ordered pair $(n, m)$ is equal to

The coefficient of ${x^2}$ in the expansion of ${\left( x - \frac{1}{2x} \right)^8}$ is

If $A$ and $B$ are coefficients of $x^{n}$ in the expansions of $(1+x)^{2n}$ and $(1+x)^{2n-1}$ respectively, then $A / B$ is equal to

$A$ possible value of $x$,for which the ninth term in the expansion of $\left\{3^{\log _{3} \sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}\right\}^{10}$ in the increasing powers of $3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}$ is equal to $180$,is:

The coefficient of $x^{12}$ in the expansion of $(x^2+2x+2)^8$ is