Given that the term of the expansion $(x^{1/3} - x^{-1/2})^{15}$ which does not contain $x$ is $5\, m$ where $m \in N$, then $m =$

The sum of the binomial coefficients of ${\left[ {2\,x\,\, + \,\,\frac{1}{x}} \right]^n}$ is equal to $256$ . The constant term in the expansion is

If the coefficients of the three consecutive terms in the expansion of $(1+ x )^{ n }$ are in the ratio $1: 5: 20$, then the coefficient of the fourth term is $............$.

If $\frac{{{T_2}}}{{{T_3}}}$ in the expansion of ${(a + b)^n}$ and $\frac{{{T_3}}}{{{T_4}}}$ in the expansion of ${(a + b)^{n + 3}}$ are equal, then $n=$

The number of positive integers $k$ such that the constant term in the binomial expansion of $\left(2 x^{3}+\frac{3}{x^{k}}\right)^{12}, x \neq 0$ is $2^{8} \cdot \ell$, where $\ell$ is an odd integer, is......

The Coefficient of $x ^{-6}$, in the expansion of $\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9$, is $........$.