In the binomial expansion of ${(a - b)^n},\,n \ge 5,$ the sum of the $5^{th}$ and $6^{th}$ terms is zero. Then $\frac{a}{b}$  is equal to

If the co-efficient of $x^9$ in $\left(\alpha x^3+\frac{1}{\beta x}\right)^{11}$ and the co-efficient of $x^{-9}$ in $\left(\alpha x-\frac{1}{\beta x^3}\right)^{11}$ are equal, then $(\alpha \beta)^2$ is equal to $.............$.

In the expansion of ${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$, the coefficient of ${x^4}$is

The number of rational terms in the binomial expansion of $\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)^{120}$ is $....$

The coefficient of $t^{50}$ in $(1 + t^2)^{25}(1 + t^{25})(1 + t^{40})(1 + t^{45})(1 + t^{47})$ is -

In the expansion of ${({5^{1/2}} + {7^{1/8}})^{1024}}$, the number of integral terms is