If the ratio of the fifth term from the begining to the fifth term from the end in the expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n$ is $\sqrt{6}: 1$, then the third term from the beginning is:
$60 \sqrt{2}$
$60 \sqrt{3}$
$30 \sqrt{2}$
$30 \sqrt{3}$
If ${x^m}$occurs in the expansion of ${\left( {x + \frac{1}{{{x^2}}}} \right)^{2n}},$ then the coefficient of ${x^m}$ is
The number of rational terms in the binomial expansion of $\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)^{120}$ is $....$
Arrange the expansion of $\left(x^{1 / 2}+\frac{1}{2 x^{1 / 4}}\right)^n$ in decreasing powers of $x$.Suppose the coeff icients 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 ${x^{100}}$ in the expansion of $\sum\limits_{j = 0}^{200} {{{(1 + x)}^j}} $ is
The coefficient of $\frac{1}{x}$ in the expansion of ${(1 + x)^n}{\left( {1 + \frac{1}{x}} \right)^n}$ is