The middle term in the expansion of ${\left( {3x - \frac{{{x^3}}}{6}} \right)^9}$ are :-

The coefficient of $x^{-5}$ in the binomial expansion of ${\left( {\frac{{x + 1}}{{{x^{\frac{2}{3}}} - {x^{\frac{1}{3}}} + 1}} - \frac{{x - 1}}{{x - {x^{\frac{1}{2}}}}}} \right)^{10}}$ where $x \ne 0, 1$ , is

If in the expansion of ${(1 + x)^{21}}$, the coefficients of ${x^r}$ and ${x^{r + 1}}$ be equal, then $r$ is equal to

The middle term in the expansion of ${(1 + x)^{2n}}$ is

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

If the coefficients of $x^7$ & $x^8$ in the expansion of ${\left[ {2\,\, + \,\,\frac{x}{3}} \right]^n}$ are equal , then the value of $n$ is :