Let $\mathrm{m}$ and $\mathrm{n}$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3} \mathrm{x}^{\frac{1}{3}}+\frac{1}{2 \mathrm{x}^{\frac{2}{3}}}\right)^{18}$. Then $\left(\frac{\mathrm{n}}{\mathrm{m}}\right)^{\frac{1}{3}}$ is :

The coefficient of $x^{1012}$ in the expansion of ${\left( {1 + {x^n} + {x^{253}}} \right)^{10}}$ , (where $n \leq 22$ is any positive integer), is

If the greatest value of the term independent of $^{\prime}x^{\prime}$ in the expansion of $\left(x \sin \alpha+a \frac{\cos \alpha}{x}\right)^{10}$ is $\frac{10 !}{(5 !)^{2}}$, then the value of $' a^{\prime}$ is equal to:

If the term independent of $x$ in the expansion of $\left(\sqrt{\mathrm{ax}}{ }^2+\frac{1}{2 \mathrm{x}^3}\right)^{10}$ is 105 , then $\mathrm{a}^2$ is equal to :

The coefficient of $x^{37}$ in the expansion of $(1-x)^{30} \, (1 + x + x^2)^{29}$ is :

If the coefficients of $(r-5)^{th}$ and $(2 r-1)^{th}$ terms in the expansion of $(1+x)^{34}$ are equal, find $r$