If the coefficient of $x$ in the expansion of ${\left( {{x^2} + \frac{k}{x}} \right)^5}$ is $270$, then $k =$

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

The natural number $m$, for which the coefficient of $x$ in the binomial expansion of $\left( x ^{ m }+\frac{1}{ x ^{2}}\right)^{22}$ is $1540,$ is

For a positive integer $n,\left(1+\frac{1}{x}\right)^{n}$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio, $2: 5: 12,$ then $n$ is equal to

If the sum of the coefficients in the expansion of $(x - 2y + 3 z)^n,$ $n \in N$ is $128$ then the greatest coefficie nt in the exp ansion of $(1 + x)^n$ is

If the third term in the binomial expansion of ${(1 + x)^m}$ is $ - \frac{1}{8}{x^2}$, then the rational value of $m$ is