The term independent of $' x '$ in the expansion of ${\left( {9\,x\,\, - \,\,\frac{1}{{3\,\sqrt x }}} \right)^{18}}, x > 0$ , is $\alpha$ times the corresponding binomial co-efficient . Then $' \alpha '$ is :

The number of integral terms in the expansion of $(7^{1/3} + 11^{1/9})^{6561}$ is :-

If $\frac{{{T_2}}}{{{T_3}}}$ in the expansion of ${(a + b)^n}$ and $\frac{{{T_3}}}{{{T_4}}}$ in the expansion of ${(a + b)^{n + 3}}$ are equal, then $n=$

Show that the coefficient of the middle term in the expansion of $(1+x)^{2 n}$ is equal to the sum of the coefficients of two middle terms in the expansion of $(1+x)^{2 n-1}$

Let $[t]$ denotes the greatest integer $\leq t$. If the constant term in the expansion of $\left(3 x^2-\frac{1}{2 x^5}\right)^7$ is $\alpha$, then $[\alpha]$ is equal to $............$.

Number of rational terms in the expansion of ${\left( {\sqrt 2 \,\, + \,\,\sqrt[4]{3}} \right)^{100}}$ is :