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 a x^2+b x+c=0 has all real roots is m n , gcd ( m , n )=1 , then m + n is equal to ________

Step-by-step solution

a , b , c 1,2,3,4 Tetrahedral dice a x^2+b x+c=0 has all real roots aligned & D 0 & b ^2-4 ac 0 aligned Let b=1 1-4 a c 0 (Not feasible) aligned & b=2 4-4 a c 0 & 1 a c a=1, c=1, & b=3 9-4 a c 0 & 9 4 a c & a=1, c=1 & a=1, c=2 & a=2, c=1 & b=4 16-4 a c 0 & 4 a c aligned 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 = 12 (4)(4)(4) = 3 16 = m m & m+n=19 aligned