In the expansion of ${\left( {x + \frac{2}{{{x^2}}}} \right)^{15}}$, the term independent of $x$ is

Sum of co-efficients of terms of degree $m$  in the expansion of $(1 + x)^n(1 + y)^n(1 + z)^n$ is

If the expansion of ${\left( {{y^2} + \frac{c}{y}} \right)^5}$, the coefficient of $y$ will be

The sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^{2}}\right)^{m}, x \neq 0, m$ being a natural number, is $559 .$ Find the term of the expansion containing $x^{3}$

In the expansion of the following expression $1 + (1 + x) + {(1 + x)^2} + ..... + {(1 + x)^n}$ the coefficient of ${x^k}(0 \le k \le n)$ is

If the ratio of the coefficient of third and fourth term in the expansion of ${\left( {x - \frac{1}{{2x}}} \right)^n}$ is $1 : 2$, then the value of $ n$ will be