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

The interval in which $x$ must lie so that the greatest term in the expansion of $(1 + x)^{2n}$ has the greatest coefficient,is

The coefficient of $x^4$ in the expansion of $\frac{1}{(1-x)(1-2x)(1-3x)}$ is

If the coefficient of $x^5$ in the expansion of $(ax^2+\frac{1}{bx})^{13}$ is equal to the coefficient of $x^{-5}$ in the expansion of $(ax-\frac{1}{bx^2})^{13}$,then $ab=$

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 $k + n_{0}$.

Arrange the expansion of $\left(x^{1/2} + \frac{1}{2x^{1/4}}\right)^n$ in decreasing powers of $x$. Suppose the coefficients of the first three terms form an arithmetic progression. Then,the number of terms in the expansion having integer power of $x$ is