Equation is $$a x^2+b x+c=0$$

$$\mathrm{D}>0$$ [for roots to be real & distinct]

$$\Rightarrow b^2-4 a c>0$$

For $$b<2$$ no value of $$a$$ & $$c$$ are possible

$$\begin{aligned} & \text { For } b=3 \Rightarrow a c<\frac{9}{4} \\ & (a, c) \in\{(1,1),(1,2),(2,1)\} \Rightarrow 3 \text { cases } \end{aligned}$$

For $$b=4 \Rightarrow a c<4$$

$$(a, c) \in\{(1,1),(1,2),(2,1),(3,1),(1,3)\} \Rightarrow 5 \text { cases }$$

For $$b=5 \Rightarrow a c<\frac{25}{4}$$

$$\begin{aligned} & (a, c) \in\{(1,1),(1,2),(2,1),(3,1),(1,3),(2,2), \\ & (4,1),(1,4),(3,2),(2,3),(5,1),(1,5),(1,6), \\ & (6,1)\}=14 \text { cases } \end{aligned}$$

For $$b=6 \Rightarrow a c<9$$

$$\begin{aligned} & (a, c) \in\{(1,1),(1,2),(2,1),(3,1),(1,3),(2,2), \\ & (4,1),(1,4),(3,2),(2,3),(5,1),(1,5),(1,6), \\ & (6,1),(2,4),(4,2)\}=16 \text { cases } \end{aligned}$$

Total cases $$=3+5+14+16=38$$ cases

$$\Rightarrow$$ Probability, $$p=\frac{38}{216}$$

$$\Rightarrow 216 p=38$$