The coefficient of ${x^{ - 9}}$ in the expansion of ${\left( {\frac{{{x^2}}}{2} - \frac{2}{x}} \right)^9}$ is

If coefficients of $2^{nd}$, $3^{rd}$ and $4^{th}$ terms in the binomial expansion of ${(1 + x)^n}$ are in $A.P.$, then ${n^2} - 9n$ is equal to

If ${\left( {2 + \frac{x}{3}} \right)^{55}}$ is expanded in the ascending powers of $x$ and the coefficients of powers of $x$ in two consecutive terms of the expansion are equal, then these terms are

If for some positive integer $n,$ the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14,$ then the largest coefficient in this expansion is

If $n$ is even positive integer, then the condition that the greatest term in the expansion of ${(1 + x)^n}$ may have the greatest coefficient also, 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