Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly \(m\) minutes. The probability that either one arrives while the other is in the cafeteria is \(40 \%,\) and \(m = a - b\sqrt {c},\) where \(a, b,\) and \(c\) are positive integers, and \(c\) is not divisible by the square of any prime. Find \(a + b + c.\)

(第十六届AIME1998 第9题)