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 ax2+bx+c=0 has all real roots is nm, gcd(m,n)=1, then m+n is equal to ______.
Answer: 19
Solution
Real roots need b2≥4ac with a,b,c∈{1,2,3,4} (64 outcomes). Counting: b=2 gives 1 way; b=3 gives 3; b=4 gives 8 (and b=1 gives none). That is 12 favourable, so probability =6412=163. With gcd(3,16)=1, m+n=3+16=19.
