If the fourth term in the binomial expansion of $\left(\sqrt{\frac{1}{x^{1+\log _{10} x}}}+x^{\frac{1}{12}}\right)^{6}$ is equal to $200$, and $x > 1$, then the value of $x$ is

The ratio of coefficient of $x^2$ to coefficient of $x^{10}$ in the expansion of ${\left( {{x^5} + {{4.3}^{ - {{\log }_{\sqrt 3 }}\sqrt {{x^3}} }}} \right)^{10}}$ is

If the ratio of the fifth term from the begining to the fifth term from the end in the expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n$ is $\sqrt{6}: 1$, then the third term from the beginning is:

The greatest term in the expansion of $\sqrt 3 {\left( {1 + \frac{1}{{\sqrt 3 }}} \right)^{20}}$ is

In the binomial expansion of ${\left( {a - b} \right)^n},n \ge 5,\;$ the sum of $5^{th}$ and $6^{th}$ terms is zero , then $a/b$ equals.

The sum of the real values of $x$ for which the middle term in the binomial expansion of ${\left( {\frac{{{x^3}}}{3} + \frac{3}{x}} \right)^8}$ equals $5670$ is