Originally Posted by

**bkbowser** 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 $\displaystyle x_i$ is the number of teachers allocated to one of the four schools then $\displaystyle x_1+x_2+x_3+x_4=8$

Comparing this too the chapter text leads me to the following, $\displaystyle \binom{n+r-1}{r-1}$, which is the number of distinct nonnegative integer-valued vectors $\displaystyle (x_1, x_2, . . . , x_r)$ satisfying the equation $\displaystyle x_1 + x_2 +...+ x_r=n$

So my first question is, since I need $\displaystyle x_i=0$, 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 $\displaystyle n=8$ and $\displaystyle r=4$, I've got $\displaystyle \binom{8+4-1}{4-1}=\frac{11!}{8!3!}=165$ 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?