counting probability with multiple cases There are four different colors of paint one can use for four different houses. If one color can be used up to three times, how many total possibilities are there? 
I approached the problem by splitting it into cases:
1) each color used once
2) one color used twice
3) two colors used twice
4) one color used thrice
but i dont know how to move on from here or if im right

How many ways could you color the four different houses without any constraints?
Call this $N$.
Once you have that figured out, consider your one constraint re-stated as:

You are not allowed to use the same color for all four houses.

Note that there are only four cases in which your constraint is violated; i.e., all four houses color 1, all four houses color 2, all four houses color 3, and all four houses color 4.
So the total will be $N - 4$.
