Number of rational terms in the expansion of ${\left( {{3^{\frac{1}{8}}} + {5^{\frac{1}{3}}}} \right)^{400}}$ is

Given that $4^{th}$ term in the expansion of ${\left( {2 + \frac{3}{8}x} \right)^{10}}$ has the maximum numerical value, the range of value of $x$ for which this will be true is given by

The sum of all those terms which are rational numbers in the expansion of $\left(2^{1 / 3}+3^{1 / 4}\right)^{12}$ is:

If for some positive integer $n,$ the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14,$ then the largest coefficient in this expansion is

The term independent of $x$ in ${\left( {\sqrt x - \frac{2}{x}} \right)^{18}}$ is

Let $\alpha>0, \beta>0$ be such that $\alpha^{3}+\beta^{2}=4 .$ If the maximum value of the term independent of $x$ in the binomial expansion of $\left(\alpha x^{\frac{1}{9}}+\beta x^{-\frac{1}{6}}\right)^{10}$ is $10 k$ then $\mathrm{k}$ is equal to