If the term independent of $x$ in the expansion of $\left( x^{\frac{2}{3}} + \frac{\alpha}{x^3} \right)^{22}$ is $7315$,then $|\alpha|$ is equal to $...........$.

If the 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

Write the general term in the expansion of $(x^{2}-yx)^{12}, x \neq 0$.

The total number of irrational terms in the binomial expansion of $(7^{1/5} - 3^{1/10})^{60}$ is:

The coefficient of $x^3$ in the expansion of $(x^2 - x + 1)^{10} (x^2 + 1)^{15}$ is equal to:

The coefficients of the $(r-1)^{th}$,$r^{th}$,and $(r+1)^{th}$ terms in the expansion of $(x+1)^{n}$ are in the ratio $1:3:5$. Find $n$ and $r$.