The constant term in the expansion of $\left(2 x+\frac{1}{x^7}+3 x^2\right)^5 \text { is }........$. 

Let the coefficients of $x ^{-1}$ and $x ^{-3}$ in the expansion of $\left(2 x^{\frac{1}{5}}-\frac{1}{x^{\frac{1}{5}}}\right)^{15}, x>0$, be $m$ and $n$ respectively. If $r$ is a positive integer such $m n^{2}={ }^{15} C _{ r } .2^{ r }$, then the value of $r$ is equal to

Find the coefficient of $x^{6} y^{3}$ in the expansion of $(x+2 y)^{9}$

If the coefficients of ${x^7}$ and ${x^8}$ in ${\left( {2 + \frac{x}{3}} \right)^n}$ are equal, then $n$ is

The largest term in the expansion of ${(3 + 2x)^{50}}$ where $x = \frac{1}{5}$ is

In the expansion of $\left(\frac{\mathrm{x}}{\cos \theta}+\frac{1}{\mathrm{x} \sin \theta}\right)^{16},$ 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