If the coefficients of the $r^{th}$ term and the $(r + 4)^{th}$ term are equal in the expansion of $(1 + x)^{20}$,then the value of $r$ is:

The term independent of $x$ in the expansion of ${\left( {{x^2} - \frac{{3\sqrt{3}}}{{{x^3}}}} \right)^{10}}$ is

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

If the second term of the expansion $\left[ a^{\frac{1}{13}} + \frac{a}{\sqrt{a^{-1}}} \right]^n$ is $14a^{5/2}$,then the value of $\frac{^nC_3}{^nC_2}$ is:

Let $a_n$ denote the term independent of $x$ in the expansion of $\left[x+\frac{\sin(1/n)}{x^2}\right]^{3n}$. Then $\lim_{n \to \infty} \frac{a_n \cdot n!}{^{3n}P_n}$ equals

Let $m$ and $n$ be the coefficients of the seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3} x^{\frac{1}{3}} + \frac{1}{2} x^{-\frac{2}{3}}\right)^{18}$. Then $\left(\frac{n}{m}\right)^{\frac{1}{3}}$ is: