The number of integral terms in the expansion of $\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680}$ is equal to

In the binomial $(2^{1/3} + 3^{-1/3})^n$, if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is $1/6$ , then $n =$

The coefficient of ${x^{53}}$ in the following expansion $\sum\limits_{m = 0}^{100} {{\,^{100}}{C_m}{{(x - 3)}^{100 - m}}} {.2^m}$is

The coefficient of ${x^{39}}$ in the expansion of ${\left( {{x^4} - \frac{1}{{{x^3}}}} \right)^{15}}$ 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.

The coefficient of $x^{4}$ is the expansion of $\left(1+\mathrm{x}+\mathrm{x}^{2}\right)^{10}$ is