The term independent of $x$ in the expansion of $\left( 9x - \frac{1}{3\sqrt{x}} \right)^{18}, x > 0$,is $\alpha$ times the corresponding binomial coefficient. Then $\alpha$ is:

The number of irrational terms in the expansion of $(3^{\frac{1}{8}}+5^{\frac{1}{4}})^{84}$ is

The term independent of $x$ in the expansion of ${\left( {\sqrt {\frac{x}{3}} + \frac{3}{{2{x^2}}}} \right)^{10}}$ is:

The expansion of $(a+x)^n$ contains $15$ terms. When $x=1$,the ratio of the neighboring terms to the middle term in this expansion is $16$. Then the positive integral value of '$a$' is

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}})^{n}$,in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6}: 1$. If the sixth term from the beginning is $\frac{\alpha}{\sqrt[4]{3}}$,then $\alpha$ is equal to $.......$

If the coefficients of the three successive terms in the binomial expansion of $(1 + x)^n$ are in the ratio $1 : 7 : 42,$ then the first of these terms in the expansion is (in $^{th}$)