In ${\left( {\sqrt[3]{2} + \frac{1}{{\sqrt[3]{3}}}} \right)^n}$ if the ratio of ${7^{th}}$ term from the beginning to the ${7^{th}}$ term from the end is $\frac{1}{6}$, then $n = $
$7$
$8$
$9$
None of these
The term independent of $x$ in the expansion of ${\left( {\frac{1}{2}{x^{1/3}} + {x^{ - 1/5}}} \right)^8}$ will be
The term independent of $x$ in the expansion of ${\left( {{x^2} - \frac{{3\sqrt 3 }}{{{x^3}}}} \right)^{10}}$ is
The coefficient of ${x^{32}}$ in the expansion of ${\left( {{x^4} - \frac{1}{{{x^3}}}} \right)^{15}}$ is
In the binomial $(2^{1/3} + 3^{-1/3})^n$, if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is $1/6$ , then $n =$
Find the middle terms in the expansions of $\left(3-\frac{x^{3}}{6}\right)^{7}$