If the middle term in the expansion of ${\left( {x^2 + \frac{1}{x}} \right)^n}$ is $924x^6$,then $n = $

Arrange the expansion of $\left(x^{1/2} + \frac{1}{2x^{1/4}}\right)^n$ in decreasing powers of $x$. Suppose the coefficients of the first three terms form an arithmetic progression. Then,the number of terms in the expansion having integer power of $x$ is

Find the term independent of $x$ in the expansion of $\left(\sqrt[3]{x}+\frac{1}{2 \sqrt[3]{x}}\right)^{18}, x > 0$.

The value of $x$,for which the $6^{th}$ 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 non-zero coefficient of the $(2r + 4)^{th}$ term is greater than the non-zero coefficient of the $(r - 2)^{th}$ term in the expansion of $(1 + x)^{18}$,then the number of possible integral values of $r$ is:

The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1 + \alpha x)^4$ and of $(1 - \alpha x)^6$ is the same if $\alpha$ equals