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

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

The coefficient of the middle term in the binomial expansion in powers of $x$ of ${(1 + \alpha x)^4}$ and of ${(1 - \alpha x)^6}$ is the same if $\alpha $ equals

$x^r$ occurs in the expansion of ${\left( {{x^3} + \frac{1}{{{x^4}}}} \right)^n}$ provided -

If the coefficient of $x ^7$ in $\left(a x-\frac{1}{b x^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(a x+\frac{1}{b x^2}\right)^{13}$ are equal, then $a^4 b^4$ is equal to :

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