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

Coefficient of $x^{11}$ in the expansion of $\left(1+x^2\right)^4\left(1+x^3\right)^7\left(1+x^4\right)^{12}$ is 

Find the coefficient of $x^{6} y^{3}$ in the expansion of $(x+2 y)^{9}$

If the term without $x$ in the expansion of $\left( x ^{\frac{2}{3}}+\frac{\alpha}{ x ^3}\right)^{22}$ is $7315$ , then $|\alpha|$ is equal to $...........$.

If the coefficients of $x^2$ and $x^3$ are both zero, in the expansion of the expression $(1 + ax + bx^2) (1 -3x)^{t5}$ in powers of $x$, then the ordered pair $(a, b)$ is equal to