The coefficients a, b, c in the quadratic equation  $a{x}^2 + bx + c = 0$ are chosen from the set $\left\{1,2,3,4,5,6,7\right\}$. The probability of this equation having repeated roots is

(1) Given Quadratic equation is $a{x}^2 + bx + c = 0$
for repeated roots $b^2 – 4ac = 0 \Rightarrow b^2 = 4ac$
$\therefore$ ac must be perfect square
$\therefore$ (a,c) $\in$ {(1,1), (1, 4), (2,2), (3,3), (4,1), (4,4), (5,5), (6,6), (7.7)}
Corresponding b must be lie in set {1,2,3,4,5,6,7}
(a,b,c) $\in$ {(1,2,1), (1,2,4), (2,4,2), (3,6,3), (4,4,1)}
Probability = $\frac{5}{7^3}$