Binomial Theorem

$\frac{1}{b}\left[\right.$ coefficient of $x^{-6}$ in $\left.\left(a x+\frac{b}{x^2}\right)^{12}\right]$

$\frac{1}{a}\left[\right.$ coefficient of $x^{-6}$ in $\left.\left(a x^2+\frac{b}{x}\right)^{12}\right]$

$a\left[\right.$ coefficient of $x^{-10}$ in $\left.\left(a x+\frac{b}{x^2}\right)^{11}\right]$

$b\left[\right.$ coefficient of $x^4$ in $\left.\left(a x^2+\frac{b}{x}\right)^{11}\right]$