The term independent of '$x$' in the expansion of $\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)^{10}$,where $x \neq 0, 1$,is equal to $.....$

The ratio of the $5^{th}$ term from the beginning to the $5^{th}$ term from the end in the binomial expansion of $\left( 2^{1/3} + \frac{1}{2(3)^{1/3}} \right)^{10}$ is

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

The coefficient of $x^{n-6}$ in the expansion $n! \left[ x - \left( \frac{^nC_0 + ^nC_1}{^nC_0} \right) \right] \left[ \frac{x}{2} - \left( \frac{^nC_1 + ^nC_2}{^nC_1} \right) \right] \left[ \frac{x}{3} - \left( \frac{^nC_2 + ^nC_3}{^nC_2} \right) \right] \dots \left[ \frac{x}{n} - \left( \frac{^nC_{n-1} + ^nC_n}{^nC_{n-1}} \right) \right]$ is equal to:

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

The coefficient of the term independent of $x$ in the expansion of $(1 + x + 2x^3) \left( \frac{3}{2}x^2 - \frac{1}{3x} \right)^9$ is