As far as "how many TOTAL coaches were fired for misconduct with players during that time span", for the purposes of this problem, it is "3".

The problem is really "What is the probability that any given 3 coaches were assistant coaches at the same school." Now, the information you need is "How many head coaches are there, total"? "How many assistant coaches are there", and "How many assistant coaches are there at each school, on average?".