Each coefficient in the equation `a x^2+b x+c=0` is determined by throwing an ordinary six faced die. Find the probability that the equation will have real roots.

Options

A. `(42)/(216)`
B. `(41)/(216)`
C. `(43)/(216)`
D. `(39)/(216)`

Answer

C

Explanation

Since each of the coefficients a, b, and c can take the values from 1 to 6.
`therefore` Total number of equations `=6xx6xx6=216`.
The roots of the equation `ax^(2)+bx+c=0` will be real if `b^(2)-4 ac ge 0 implies b^(2) ge 4` ac.
The favourable number of elementary events can be enumerated as follows :

Since `b^(2) ge 4 ac` and since the maximum value of `b^(2)` is 36, therefore ac=10,11,12.. etc. is not possible.
`therefore` Total number of favourable elementary events =43
Hence, required probability `=(43)/(216)`