Question 9
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 $a + b + c.$