In the binomial expansion of $(1+x)^{15}$,the coefficients of $x^{r}$ and $x^{r+3}$ are equal. Then,$r$ is

If ${x^m}$ occurs in the expansion of ${\left( {x + \frac{1}{{{x^2}}}} \right)^{2n}},$ then the coefficient of ${x^m}$ is

The coefficients of three consecutive terms of $(1+x)^{n+5}$ are in the ratio $5: 10: 14$. Then $n=$

The middle terms in the expansion of ${\left( {3x - \frac{{{x^3}}}{6}} \right)^9}$ are:

The number of irrational terms in the binomial expansion of $(3^{1/5} + 7^{1/3})^{100}$ 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