JEE Main 2024 — Mathematics Probability & Statistics

$\mathrm{a}, \mathrm{b}, \mathrm{c} \in\{1,2,3,4\}$a,b,c∈{1,2,3,4}

Tetrahedral dice $a x^2+b x+c=0$ax2+bx+c=0 has all real roots $\begin{aligned}
& \Rightarrow \mathrm{D} \geq 0 \
& \Rightarrow \mathrm{b}^2-4 \mathrm{ac} \geq 0
\end{aligned}$

Let $b=1 \Rightarrow 1-4 a c \geq 0$b=1⇒1−4ac≥0 (Not feasible) $\begin{aligned} & b=2 \Rightarrow 4-4 a c \geq 0 \\ & 1 \geq a c \Rightarrow a=1, c=1, \\ & b=3 \Rightarrow 9-4 a c \geq 0 \\ & \frac{9}{4} \geq a c \\ & \Rightarrow a=1, c=1 \\ & \Rightarrow a=1, c=2 \\ & \Rightarrow a=2, c=1 \\ & b=4 \Rightarrow 16-4 a c \geq 0 \\ & 4 \geq a c\end{aligned}$​b=2⇒4−4ac≥01≥ac⇒a=1,c=1,b=3⇒9−4ac≥049​≥ac⇒a=1,c=1⇒a=1,c=2⇒a=2,c=1b=4⇒16−4ac≥04≥ac​ $\begin{aligned} & \Rightarrow a=1, c=1 \\ & \Rightarrow a=1, c=2 \quad \Rightarrow a=2, c=1 \\ & \Rightarrow a=1, c=3 \quad \Rightarrow a=3, c=1 \\ & \Rightarrow a=1, c=4 \quad \Rightarrow a=4, c=1 \\ & \Rightarrow a=2, c=2 \\ & \text { Probability }=\frac{12}{(4)(4)(4)}=\frac{3}{16}=\frac{m}{m} \\ & m+n=19\end{aligned}$​⇒a=1,c=1⇒a=1,c=2⇒a=2,c=1⇒a=1,c=3⇒a=3,c=1⇒a=1,c=4⇒a=4,c=1⇒a=2,c=2 Probability =(4)(4)(4)12​=163​=mm​m+n=19​