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

The coefficient of $x^5$ in the expansion of $\left(2 x^3-\frac{1}{3 x^2}\right)^5$ is

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

The coefficient of $t^{12}$ in the expansion of $(1 + t^2)^6(1 + t^6)(1 + t^{12})$ is:

Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x)^{99}$. Let $a$ be the middle term in the expansion of $(2+\frac{1}{\sqrt{2}})^{200}$. If $\frac{{}^{200}C_{99} K}{a} = \frac{2^{\ell} m}{n}$,where $m$ and $n$ are odd numbers,then the ordered pair $(\ell, n)$ is equal to:

Let the $9^{\text{th}}$ term in the binomial expansion of $(3+6x)^{n}$,in the increasing powers of $6x$,be the greatest for $x=\frac{3}{2}$. If $n_{0}$ is the least value of $n$ for which this holds,and $k$ is the ratio of the coefficient of $x^{6}$ to the coefficient of $x^{3}$,then find the value of $k + n_{0}$.