If in the expansion of $(1 + x)^{21}$,the coefficients of $x^r$ and $x^{r + 1}$ are equal,then $r$ is equal to

In the binomial $(2^{1/3} + 3^{-1/3})^n$,if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is $1/6$,then $n =$

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}})^{n}$,in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6}: 1$. If the sixth term from the beginning is $\frac{\alpha}{\sqrt[4]{3}}$,then $\alpha$ is equal to $.......$

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

The absolute value of the numerically greatest term in the expansion of $(2x - 3y)^{12}$ when $x = 3$ and $y = 2$ is:

The coefficient of $t^4$ in the expansion of ${\left( {\frac{{1 - {t^6}}}{{1 - t}}} \right)^3}$ is