The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in the expansion of $(1+x)^{19}$ is $....$

Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{x} - \frac{6}{x^{3/2}}\right)^n$,$n \leq 15$. If the sum of the coefficients of the remaining terms in the expansion is $649$ and the coefficient of $x^{-n}$ is $\lambda \alpha$,then $\lambda$ is equal to $..........$.

If $n$ is even and the middle term in the expansion of $\left(x^2+\frac{1}{x}\right)^n$ is $924 x^6$,then $n$ is equal to

The coefficient of $x^8$ in the expansion of $(1 - x^4)^4 (1 + x)^5$ is :-

The coefficient of $x^{-7}$ in the expansion of $\left( ax - \frac{1}{bx^2} \right)^{11}$ is

The term independent of $x(x>0, x \neq 1)$ in the expansion of $\left[\frac{(x+1)}{\left(x^{2 / 3}-x^{1 / 3}+1\right)}-\frac{(x-1)}{(x-\sqrt{x})}\right]^{10}$ is: