If the non zero coefficient of $(2r + 4)th$ term is greater than non zero coefficient of $(r - 2)th$ term in expansion of $(1 + x)^{18}$, then number of possible integral values of $r$ is

If some three consecutive in the binomial expansion of ${\left( {x + 1} \right)^n}$ in powers of $x$ are in the ratio $2 : 15 : 70$, then the average of these three coefficient is

Find $n,$ if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{n}$ is $\sqrt{6}: 1$

The coefficient of ${x^7}$ in the expansion of ${\left( {\frac{{{x^2}}}{2} - \frac{2}{x}} \right)^8}$ is

In the expansion of ${\left( {\frac{a}{x} + bx} \right)^{12}}$,the coefficient of $x^{-10}$ will be

The positive value of $a$ so that the co-efficient of $x^5$ is equal to that of $x^{15}$ in the expansion of ${\left( {{x^2}\,\, + \,\,\frac{a}{{{x^3}}}} \right)^{10}}$ is