If 8 identical blackboards are to be divided among 4 schools, how many divisions are possible? how many if each school must receive at least 1 blackboard?

Alright, so we'd use the combinations with repetition formula, so we choose from 4 schools to distribute to and distribute 8 blackboards. It's then

(8+4-1) !/8! (4-1) !=11! / (3!*8!) = 165

For at least one blackboard, we first distribute 1 to each school and then have 4 blackboards left, getting (4+4-1) !/4! (4-1) !=7! / (4!*3!) = 35