The term independent of $x$ in the expansion ${\left( {{x^2} - \frac{1}{{3x}}} \right)^9}$ is

In the expansion of ${\left( {3x - \frac{1}{{{x^2}}}} \right)^{10}}$ then $5^{th}$ term from the end is :-

Let the coefficients of $x ^{-1}$ and $x ^{-3}$ in the expansion of $\left(2 x^{\frac{1}{5}}-\frac{1}{x^{\frac{1}{5}}}\right)^{15}, x>0$, be $m$ and $n$ respectively. If $r$ is a positive integer such $m n^{2}={ }^{15} C _{ r } .2^{ r }$, then the value of $r$ is equal to

In the expansion of $(1+a)^{m+n},$ prove that coefficients of $a^{m}$ and $a^{n}$ are equal.

Find $a$ if the coefficients of $x^{2}$ and $x^{3}$ in the expansion of $(3+a x)^{9}$ are equal.

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