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

The term independent of $x$ in the expansion of ${\left( {{x^2} - \frac{{3\sqrt 3 }}{{{x^3}}}} \right)^{10}}$ is

Find the $13^{\text {th }}$ term in the expansion of $\left(9 x-\frac{1}{3 \sqrt{x}}\right)^{18}, x \neq 0$

If the term without $x$ in the expansion of $\left( x ^{\frac{2}{3}}+\frac{\alpha}{ x ^3}\right)^{22}$ is $7315$ , then $|\alpha|$ is equal to $...........$.

If the coefficients of ${5^{th}}$, ${6^{th}}$and ${7^{th}}$ terms in the expansion of ${(1 + x)^n}$be in $A.P.$, then $n =$