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 = $

If the constant term in the expansion of $\left(1+2 x-3 x^3\right)\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9$ is $p$, then $108$ p is equal to………………..

In the binomial expansion of ${(a – b)^n},\,n \ge 5,$ the sum of the $5^{th}$ and $6^{th}$ terms is zero. Then $\frac{a}{b}$  is equal to

The term independent of $x$ in expansion of ${\left( {\frac{{x + 1}}{{{x^{2/3}} – {x^{\frac{1}{3}}} + 1\;}}–\frac{{x – 1}}{{x – {x^{1/2}}}}} \right)^{10}}$ is

The interval in which $x$ must lie so that the greatest term in the expansion of ${(1 + x)^{2n}}$ has the greatest coefficient, is