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

If in the expansion of $(a-2b)^{n}$, the sum of the $5^{th}$ and $6^{th}$ term is zero, then the value of $\frac{a}{b}$ is

The coefficient of $x^{256}$ in the expansion of $(1-x)^{101}(x^{2}+x+1)^{100}$ is:

Let $\alpha > 0, \beta > 0$ be such that $\alpha^{3} + \beta^{2} = 4$. If the maximum value of the term independent of $x$ in the binomial expansion of $(\alpha x^{\frac{1}{9}} + \beta x^{-\frac{1}{6}})^{10}$ is $10k$,then $k$ is equal to

If the $(r + 1)^{th}$ term in the expansion of $\left( \sqrt[3]{\frac{a}{\sqrt{b}}} + \sqrt{\frac{b}{\sqrt[3]{a}}} \right)^{21}$ has the same power of $a$ and $b$,then the value of $r$ is

For a positive integer $n$,$(1+\frac{1}{x})^n$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio $2:5:12$,then $n$ is equal to