If the coefficient of the middle term in the expansion of $(1 + x)^{2n + 2}$ is $p$ and the coefficients of the two middle terms in the expansion of $(1 + x)^{2n + 1}$ are $q$ and $r$,then:

If ${x^m}$ occurs in the expansion of ${\left( {x + \frac{1}{{{x^2}}}} \right)^{2n}},$ then the coefficient of ${x^m}$ is

The number of rational terms in the binomial expansion of $(\sqrt[4]{5}+\sqrt[5]{4})^{100}$ 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 sum of the rational terms in the binomial expansion of $(\sqrt{2} + 3^{1/5})^{10}$ is

If the coefficients of $x^{7}$ in $(x^{2}+\frac{1}{bx})^{11}$ and $x^{-7}$ in $(x-\frac{1}{bx^{2}})^{11}$,$b \neq 0$,are equal,then the value of $b$ is equal to: