If the coefficients of ${5^{th}}$, ${6^{th}}$and ${7^{th}}$ terms in the expansion of ${(1 + x)^n}$be in $A.P.$, then $n =$

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 = $

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

The number of terms in the expansion of  ${\left( {\sqrt[4]{9} + \sqrt[6]{8}} \right)^{500}}$, which are integers is

The coefficient of the term independent of $x$ in the expansion of $(1 + x + 2x^3)$ ${\left( {\frac{3}{2}{x^2} - \frac{1}{{3x}}} \right)^9}$ is

If the coefficient of $4^{th}$ term in the expansion of ${(a + b)^n}$ is $56$, then $n$ is