Question

Let a, b and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4. If the probability that \(\text{ax}^2 \) + bx + c = 0 has all real roots is \(\frac{m}{n}\) , gcd(m, n) = 1, then m + n is equal to ________.

Answer 1

Correct answer is : 19 

\(a, b, c \in\{1,2,3,4\}\) 

Tetrahedral dice

\(a x^{2}+b x+c=0\)

has all real roots

\(\Rightarrow \mathrm{D} \geq 0\)

\(\Rightarrow \mathrm{b}^{2}-4 \mathrm{ac} \geq 0\)

Let \( \mathrm{b}=1 \Rightarrow 1-4 \mathrm{ac} \geq 0\) (Not feasible)

\(\mathrm{b}=2 \Rightarrow 4-4 \mathrm{ac} \geq 0\)

\(1 \geq \mathrm{ac} \Rightarrow \mathrm{a}=1, \mathrm{c}=1\),

\(\mathrm{b}=3 \Rightarrow 9-4 \mathrm{ac} \geq 0\)

\(\frac{9}{4} \geq\) ac

\(\Rightarrow \mathrm{a}=1, \mathrm{c}=1\)

\(\Rightarrow \mathrm{a}=1, \mathrm{c}=2\)

\(\Rightarrow \mathrm{a}=2, \mathrm{c}=1\)

\(\mathrm{b}=4 \Rightarrow 16-4 \mathrm{ac} \geq 0\)

\(4 \geq\) ac

\(\Rightarrow \mathrm{a}=1, \mathrm{c}=1\)

\(\Rightarrow \mathrm{a}=1, \mathrm{c}=2 \quad \Rightarrow \mathrm{a}=2, \mathrm{c}=1\)

\(\Rightarrow \mathrm{a}=1, \mathrm{c}=3 \quad \Rightarrow \mathrm{a}=3, \mathrm{c}=1\)

\(\Rightarrow \mathrm{a}=1, \mathrm{c}=4 \quad \Rightarrow \mathrm{a}=4, \mathrm{c}=1\)

\(\Rightarrow \mathrm{a}=2, \mathrm{c}=2\)

Probability \(=\frac{12}{(4)(4)(4)}=\frac{3}{16}=\frac{\mathrm{m}}{\mathrm{m}}\)

\(\mathrm{m}+\mathrm{n}=19\)