Three numbers a, b and c are drawn independently with replacement from the set 1, 2, 3, 4, 5 . If the probability that the line ax + by + c = 0 intersects the circle x^2 + y^2 = 1 at two distinct points is m n , where (m, n) = 1 , then m + n is equal to ________

Step-by-step solution

For the line ax + by + c = 0 to intersect the circle x^2 + y^2 = 1 at two distinct points, the perpendicular distance from the centre (0, 0) to the line must be strictly less than the radius 1 . |c| a^2 + b^2 The total number of possible outcomes for (a, b, c) is 5 5 5 = 125 . We count the favourable outcomes by considering each possible value of c : If c = 1 , a^2 + b^2 > 1 . This is true for all 25 pairs of (a, b) . If c = 2 , a^2 + b^2 > 4 . The only pair that fails is (1, 1) . Favourable pairs = 25 - 1 = 24 . I