If 8 new teachers are to be divided among 4 schools, how many divisions are possible? What if each school must receive 2 teachers?
For the first question, by stars and bars, there are C(11,3) = 165 ways.
For the second question, if each school must receive 2 teachers, there are 4! = 24 possible divisions.
The reason for the difference is that in the first case, the teachers can be divided among the schools in any way. In the second case, the teachers must be divided in such a way that each school receives 2 teachers. This limits the number of possible divisions.