If the $6^{th}$ term in the expansion of the binomial ${\left[ {\sqrt {{2^{\log (10 - {3^x})}}} + \sqrt[5]{{{2^{(x - 2)\log 3}}}}} \right]^m}$ is equal to $21$ and it is known that the binomial coefficients of the $2^{nd}$, $3^{rd}$ and $4^{th}$ terms in the expansion represent respectively the first, third and fifth terms of an $A.P$. (the symbol log stands for logarithm to the base $10$), then $x = $

Let for the $9^{\text {th }}$ term in the binomial expansion of $(3+6 x)^{n}$, in the increasing powers of $6 x$, to be the greatest for $x=\frac{3}{2}$, the least value of $n$ is $n_{0}$. If $k$ is the ratio of the coefficient of $x ^{6}$ to the coefficient of $x ^{3}$, then $k + n _{0}$ is equal to.

${16^{th}}$ term in the expansion of ${(\sqrt x - \sqrt y )^{17}}$ is

Find the $r^{\text {th }}$ term from the end in the expansion of $(x+a)^{n}$

In the expansion of ${\left( {2{x^2} - \frac{1}{x}} \right)^{12}}$, the term independent of x is

Find the middle terms in the expansions of $\left(3-\frac{x^{3}}{6}\right)^{7}$