Question

If coefficients $a,b,c$ of quadratic equation $a{x}^2+bx+c=0$ are chosen at random with replacement from the set $S=1,2,3,4,5,6$, find out the probability that roots of quadratic are real and distinct.

• A. $\frac{20}{108}$
• B. $\frac{19}{108}$
• C. $\frac{21}{108}$
• D. $\frac{22}{108}$

Answer 1

The correct option is B $\frac{19}{108}$

For real and distinct solution: $b^2>4ac$

$a$	$(b,c)$	Total
$1$	-	$0$
$2$	-	$0$
$3$	$(1,1),(1,2),(2,1)$	$3$
$4$	$(1,1),(1,2),(2,1),(1,3),(3,1)$	$5$
$5$	$(1,1),(1,2),(2,1),(1,3),(3,1),(2,2),(2,3),(3,2),(1,4),(4,1),(1,5),(5,1),(1,6),(6,1)$	$14$
$6$	$(1,1),(1,2),(2,1),(1,3),(3,1),(2,2),(2,3),(3,2),(1,4),(4,1),(1,5),(5,1),(1,6),(6,1),(2,4),(4,2)$	$16$
	Total	$38$

Total no. of ways of choosing $a,b,c=6^3=216$
Required probability,

$P=\frac{38}{216}=\frac{19}{108}$

Hence, option B.