The coefficient of $x^4$ in the expansion of $(1-x+x^2-x^3)^4$ is

Let the $9^{\text{th}}$ term in the binomial expansion of $(3+6x)^{n}$,in the increasing powers of $6x$,be the greatest for $x=\frac{3}{2}$. If $n_{0}$ is the least value of $n$ for which this holds,and $k$ is the ratio of the coefficient of $x^{6}$ to the coefficient of $x^{3}$,then find the value of $k + n_{0}$.

The term independent of $y$ in the expansion of $(y^{-1/6} - y^{1/3})^9$ is

The greatest term in the expansion of $\sqrt{3} \left( 1 + \frac{1}{\sqrt{3}} \right)^{20}$ is

Find the $r^{\text{th}}$ term from the end in the expansion of $(x+a)^{n}$.

If the coefficients of the $(2r + 1)^{th}$ term and the $(r + 2)^{th}$ term are equal in the expansion of $(1 + x)^{43}$,then the value of $r$ is: