In the expansion of ${\left( {\frac{{x\,\, + \,\,1}}{{{x^{\frac{2}{3}}}\,\, - \,\,{x^{\frac{1}{3}}}\,\, + \,\,1}}\,\, - \,\,\frac{{x\,\, - \,\,1}}{{x\,\, - \,\,{x^{\frac{1}{2}}}}}} \right)^{10}}$, the term which does not contain $x$ is :

Find the middle terms in the expansion of $\left(\frac{x}{3}+9 y\right)^{10}$

Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{ x }-\frac{6}{ x ^{\frac{3}{2}}}\right)^{ n }, n \leq 15$. 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 for positive integers $r > 1,n > 2$ the coefficient of the ${(3r)^{th}}$ and ${(r + 2)^{th}}$ powers of $x$ in the expansion of ${(1 + x)^{2n}}$ are equal, then

${16^{th}}$ term in the expansion of ${(\sqrt x - \sqrt y )^{17}}$ is

In the expansion of $(1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0$, the sum of the coefficient of $x^3$ and $x^{-13}$ is equal to