The sum of the real values of $x$ for which the middle term in the binomial expansion of ${\left( {\frac{{{x^3}}}{3} + \frac{3}{x}} \right)^8}$ equals $5670$ is

If the constant term in the binomial expansion of $\left(\frac{x^{5/2}}{2} - \frac{4}{x^{\ell}}\right)^9$ is $-84$ and the coefficient of $x^{-3\ell}$ is $2^{\alpha}\beta$,where $\beta < 0$ is an odd number,then $|\alpha\ell - \beta|$ is equal to

The coefficient of $t^{50}$ in $(1 + t^2)^{25}(1 + t^{25})(1 + t^{40})(1 + t^{45})(1 + t^{47})$ is -

The coefficient of $x^{4}$ in the expansion of $(1+x+x^{2}+x^{3})^{6}$ in powers of $x$ is:

Find the middle term of the expansion of $ \left(\frac{10}{x}+\frac{x}{10}\right)^{10} $.

If the $7^{th}$ term in the binomial expansion of $\left( \frac{3}{\sqrt[3]{84}} + \sqrt{3} \ln x \right)^9, x > 0,$ is equal to $729,$ then $x$ can be