Mathematics - Probability Question with Solution | TestHub

$\mathrm{a},\mathrm{b},\mathrm{c}\in \{1,2,3,4\}$
Tetrahedral dice $a{x}^2+bx+c=0$ has all real roots $\begin{array}{rl}& \Rightarrow \mathrm{D}\ge 0\\ & \Rightarrow {\mathrm{b}}^2-4\mathrm{ac}\ge 0\end{array}$
Let $b=1\Rightarrow 1-4ac\ge 0$ (Not feasible) $\begin{array}{rl}& b=2\Rightarrow 4-4ac\ge 0\\ & 1\ge ac\Rightarrow a=1,c=1,\\ & b=3\Rightarrow 9-4ac\ge 0\\ & \frac{9}{4}\ge ac\\ & \Rightarrow a=1,c=1\\ & \Rightarrow a=1,c=2\\ & \Rightarrow a=2,c=1\\ & b=4\Rightarrow 16-4ac\ge 0\\ & 4\ge ac\end{array}$ $\begin{array}{rl}& \Rightarrow a=1,c=1\\ & \Rightarrow a=1,c=2\Rightarrow a=2,c=1\\ & \Rightarrow a=1,c=3\Rightarrow a=3,c=1\\ & \Rightarrow a=1,c=4\Rightarrow a=4,c=1\\ & \Rightarrow a=2,c=2\\ & \text{ Probability }=\frac{12}{(4)(4)(4)}=\frac{3}{16}=\frac{m}{m}\\ & m+n=19\end{array}$