Each coefficient in the equation $a x^{2} + b x + c = 0$ is determined by throwing an ordinary die. The probability that the equation will have equal roots is $k / 216$ . Find the value of $k$ .

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Solution

Roots equal $\Rightarrow b^{2} - 4 a c = 0 \Rightarrow {(\frac{b}{2})}^{2} = a c$
Each coefficients is an integer, So we consider the following cases
$b = 1 \Rightarrow \frac{1}{4} = a c$ No integral value of $a$ and $c$
$b = 2 \Rightarrow 1 = a c ∴ (1, 1)$
$b = 3 \Rightarrow \frac{9}{2} = a c$ No integral value of $a$ and $c$
$b = 4 \Rightarrow 4 = a c ∴ (1, 4), (2, 2), (4, 1)$
$b = 5 \Rightarrow \frac{25}{2} = a c$ No integral value of $a$ and $c$
$b = 6 \Rightarrow 9 = a c ∴ (3, 3)$
Thus we have $5$ favorable ways for $b = 2, 4, 6$
Total number of equation is $6.6.6 = 216$
Therefore required probability is $\frac{5}{216}$

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