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

Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2+(1+x)^3+\cdots+(1+x)^{49}+(1+mx)^{50}$ is $(3n+1)^{51}C_3$ for some positive integer $n$. Then the value of $n$ is

The coefficient of $x^{n-6}$ in the expansion $n! \left[ x - \left( \frac{^nC_0 + ^nC_1}{^nC_0} \right) \right] \left[ \frac{x}{2} - \left( \frac{^nC_1 + ^nC_2}{^nC_1} \right) \right] \left[ \frac{x}{3} - \left( \frac{^nC_2 + ^nC_3}{^nC_2} \right) \right] \dots \left[ \frac{x}{n} - \left( \frac{^nC_{n-1} + ^nC_n}{^nC_{n-1}} \right) \right]$ is equal to:

The term independent of $x$ in the expansion of ${(1 + x)^n}{\left( {1 + \frac{1}{x}} \right)^n}$ is

Find the term independent of $x$ $(x > 0, x \neq 1)$ in the expansion of $\left[\frac{(x+1)}{\left(x^{2/3}-x^{1/3}+1\right)}-\frac{(x-1)}{(x-\sqrt{x})}\right]^{10}$.

The coefficient of $x^2$ in the expansion of $(2x^2 + \frac{1}{x})^{10}, x \neq 0$, is: