If the coefficients of ${x^7}$ and ${x^8}$ in ${\left( {2 + \frac{x}{3}} \right)^n}$ are equal, then $n$ is

The value of $x$, for which the 6th term in the expansion of ${\left\{ {{2^{{{\log }_2}\sqrt {({9^{x - 1}} + 7)} }} + \frac{1}{{{2^{(1/5){{\log }_2}({3^{x - 1}} + 1)}}}}} \right\}^7}$ is $84$, is equal to

If the coefficients of ${T_r},\,{T_{r + 1}},\,{T_{r + 2}}$ terms of ${(1 + x)^{14}}$ are in $A.P.$, then $r =$

The greatest coefficient in the expansion of ${(1 + x)^{2n + 1}}$ is

Coefficient of ${t^{12}}$  in ${\left( {1 + {t^2}} \right)^6}\left( {1 + {t^6}} \right)\left( {1 + {t^{12}}} \right)$ is-

The middle term in the expression of ${\left( {x - \frac{1}{x}} \right)^{18}}$ is