The total number of irrational terms in the binomial expansion of $(7^{1/5} - 3^{1/10})^{60}$ is:

If $a>0$ and the coefficient of $x^2$ in the expansion of $\left(a x^3+\frac{c}{x}\right)^6$ is $60$,then $a c^2=$

The maximum value of the term independent of $t$ in the expansion of $\left( tx^{\frac{1}{5}} + \frac{(1-x)^{\frac{1}{10}}}{t} \right)^{10}$ where $x \in (0, 1)$ is

The coefficient of $x^7$ in the expansion of $(1 - x - x^2 + x^3)^6$ is

If the numerically greatest term in the expansion of $(2-3x)^9$ when $x=1$ is $P_1^\alpha P_2^\beta P_3^\gamma P_4^\delta$ (where $P_1 < P_2 < P_3 < P_4$ are the first four prime numbers),then $\alpha+\beta+\gamma+\delta=$

In the expansion of $\left(\frac{x}{\cos \theta}+\frac{1}{x \sin \theta}\right)^{16},$ if $\ell_{1}$ is the least value of the term independent of $x$ when $\frac{\pi}{8} \leq \theta \leq \frac{\pi}{4}$ and $\ell_{2}$ is the least value of the term independent of $x$ when $\frac{\pi}{16} \leq \theta \leq \frac{\pi}{8},$ then the ratio $\ell_{2} : \ell_{1}$ is equal to