If 8 new teachers are to be divided among 4 schools, how many divisions are possible?
I understand that in this question you are just solving for the multinomial coefficients of the multinomial theorem. So x1=x2=x3=x4=1:
(1+1+1+1)^8 = 65,536.
But I thought about this question a different way and don't understand why this solution is not working:
Suppose I have 8 teachers: A B C D E F G H and the '^' symbol represents the dividing line for the school. Since I have 4 schools, I will have 3 dividers.
So an example division would be as such: AB^CD^EF^GH. In this case,
Another example would be ABC^D^EF^GH:
So how many different ways could I place the divisors? The answer is 8-1 and I have to choose 3. Also there are 8! permutations of the teachers. So I thought if I do: (7 choose 3) * 8! I would get the answer, but I don't. Can someone please explain what I am misunderstanding and what I just actually calculated? Thanks in advance!