An old billionaire is writing his will, and deciding how to distribute his nine mansions among his three children. He wants to give each child at least one mansion, and he does not want to give any two of his children the same number of mansions. In how many possible ways can he distribute his mansions?
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. :)