If the fourth term in the Binomial expansion of ${\left( {\frac{2}{x} + {x^{\log_8 x}}} \right)^6}$ for $x > 0$ is $20 \times 8^7$,then a value of $x$ is

Let $(1+x+2x^2)^{20} = a_0 + a_1x + a_2x^2 + \ldots + a_{40}x^{40}$,then $a_1 + a_3 + a_5 + \ldots + a_{37}$ is equal to

In the expansion of ${\left( {3x - \frac{1}{{{x^2}}}} \right)^{10}}$,the $5^{th}$ term from the end is:

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

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

The coefficient of $x^{24}$ in the expansion of $(1+x^2)^{12}(1+x^{12})(1+x^{24})$ is