If the fourth term in the Binomial expansion of ${\left( {\frac{2}{x} + {x^{{{\log }_e}x}}} \right)^6}(x > 0)$ is $20\times 8^7,$ then a value of $x$ is
$8^3$
$8^{-2}$
$8$
$8^2$
Find the middle terms in the expansion of $\left(\frac{x}{3}+9 y\right)^{10}$
Suppose $2-p, p, 2-\alpha, \alpha$ are the coefficient of four consecutive terms in the expansion of $(1+x)^n$. Then the value of $p^2-\alpha^2+6 \alpha+2 p$ equals
The coefficients of three consecutive terms of $(1+x)^{n+5}$ are in the ratio $5: 10: 14$. Then $n=$
If the coefficient of $x$ in the expansion of ${\left( {{x^2} + \frac{k}{x}} \right)^5}$ is $270$, then $k =$
The interval in which $x$ must lie so that the greatest term in the expansion of ${(1 + x)^{2n}}$ has the greatest coefficient, is