You can start by giving one mansion to each child ahead of time. Now you have to distribute 6 mansions among three children. You use stars and bars and get 9 choose 3, which is 84 possibilities. Now you have to subtract the number of possibilities in which the two or three children have the same houses.

Everyone gets 3 houses: 1 possibility.

Two people get 1 house: 3 possibilities. This is because A and B get them, B and C get them, or A and C get them.

Two people get 2 houses: 3 possibilities.

Two people get 4 houses: 3 possibilities.

84-3-3-3-1 = 74.

**The Answer is 74 possibilities.**

Hope I helped. :)