If values of a,b,c is given from the set {1,2,3} where a not equal to b not equal to c, then the probability that b²-4ac>0 is what?

Dear Student,

as a is not equal to b and b is not equal to c.
So we can select a,b,c from set of {1,2,3} such that numbers are not repeated.
So this can be done in 3! ways.

This is the total sample size= 6.
Now to satisfy b² -4ac>0

Take b=3, a=1,c=2
$$\begin{gathered} 3^2-4\times 1\times 2 \\ 9-8=1>0 \\ Hence (a,b,c)=(1,3,2) satisfies the condition \\ let b=3, a=2,c=1 \\ {b}^2-4ac>0 \\ {3}^2-4\times 1\times 2 \\ 9-8=1>0 \\ thus (a,b,c)=(2,3,1) satisfies the condition \\ take b=2 \\ {2}^2-4\times 1\times 3 \\ 4-12<0 \\ equation not satisfied \\ also check for b=1 \\ {1}^2-4\times 2\times 3 \\ 1-24<0 \\ Hence only two combination (2,3,1),(1,3,2) satisfy the given condition. \\ Thus favourable cases=2 \\ Probability that a,b,c satisfy b^2-4ac>0 \\ P=\frac{2}{6} \\ =\frac{1}{3} \end{gathered}$$
Regards,