The positive value of $\lambda $ for which the co-efficient of $x^2$ in the expression ${x^2}{\left( {\sqrt x  + \frac{\lambda }{{{x^2}}}} \right)^{10}}$ is $720$ is

The middle term in the expansion of ${\left( {x + \frac{1}{x}} \right)^{10}}$ is

If the coefficients of $x$ and $x^{2}$ in the expansion of $(1+x)^{p}(1-x)^{q}, p, q \leq 15$, are $-3$ and $-5$ respectively, then the coefficient of $x ^{3}$ is equal to $............$

The coefficients of three successive terms in the expansion of ${(1 + x)^n}$ are $165, 330$ and $462$ respectively, then the value of n will be

Write the general term in the expansion of $\left(x^{2}-y\right)^{6}$

The coefficient of $x^{18}$ in the expansion of $\left(x^4-\frac{1}{x^3}\right)^{15}$ is $...........$.