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

If the third term in the binomial expansion of ${\left( {1 + {x^{{{\log }_2}\,x}}} \right)^5}$ equals $2560$, then a possible value of $x$ is

The interval in which $x$ must lie so that the greatest term in the expansion of ${(1 + x)^{2n}}$ has the greatest coefficient, is

In the binomial expansion of ${\left( {a - b} \right)^n},n \ge 5,\;$ the sum of $5^{th}$ and $6^{th}$ terms is zero , then $a/b$ equals.

Number of integral tems in the expansion of  $\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}^{824}$  is equal to..................

If the second, third and fourth terms in the expansion of $(x+y)^{\mathrm{n}}$ are $135$,$30$ and $\frac{10}{3}$, respectively, then $6\left(n^3+x^2+y\right)$ is equal to .............