The ratio of coefficient of $x^2$ to coefficient of $x^{10}$ in the expansion of ${\left( {{x^5} + {{4.3}^{ - {{\log }_{\sqrt 3 }}\sqrt {{x^3}} }}} \right)^{10}}$ is

If the coefficient of $x ^{10}$ in the binomial expansion of $\left(\frac{\sqrt{x}}{5^{\frac{1}{4}}}+\frac{\sqrt{5}}{x^{\frac{1}{3}}}\right)^{60}$ is $5^{ k } l$, where $l, k \in N$ and $l$ is coprime to $5$ , then $k$ is equal to

In the expansion of ${(1 + x)^n}$ the coefficient of $p^{th}$ and ${(p + 1)^{th}}$ terms are respectively $p$ and $q$. Then $p + q = $

If ${\left( {2 + \frac{x}{3}} \right)^{55}}$ is expanded in the ascending powers of $x$ and the coefficients of powers of $x$ in two consecutive terms of the expansion are equal, then these terms are

The sum of the rational terms in the binomial expansion of ${\left( {{2^{\frac{1}{2}}} + {3^{\frac{1}{5}}}} \right)^{10}}$ is

The coefficients of three successive terms in the expansion of ${(1 + x)^n}$ are $165, 330$ and $462$ respectively, then the value of n will be