The coefficient of $x^{11}$ in the expansion of $(1+x^2)^4(1+x^3)^7(1+x^4)^{12}$ is:

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

The coefficient of $x^{10}$ in the expansion of $1+(1+x)+\dots+(1+x)^{20}$

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:

If the sum of the coefficients of all the positive even powers of $x$ in the binomial expansion of $(2x^3 + \frac{3}{x})^{10}$ is $5^{10} - \beta \cdot 3^9$,then $\beta$ is equal to

In ${\left( {\sqrt[3]{2} + \frac{1}{{\sqrt[3]{3}}}} \right)^n}$,if the ratio of the $7^{th}$ term from the beginning to the $7^{th}$ term from the end is $\frac{1}{6}$,then $n = $