28. If 8 new teachers are to be divided among 4 schools, how many divisions are possible? What if each school must receive 2 teachers?

OK, so it's pretty clear that if

is the number of teachers allocated to one of the four schools then

Comparing this too the chapter text leads me to the following,

, which is the number of distinct nonnegative integer-valued vectors

satisfying the equation

So my first question is, since I need

, to be a valid entry does this formula cover that case? (I'm thinking it does because it mentions nonnegative integer-valued vectors.)

Secondly, taking

and

, I've got

as my current working solution. However, the text is talking about indistinguishable objects here and I hardly think teachers are indistinguishable. Also, the listed answer is 65,536. So I think I need to permute the teachers now somehow? Or is this all wrong?