Distributing balls in boxes There are five balls of identical sizes but different colors. One of the balls is red, one is blue, one is green and the other two are yellow. Moreover, there are three boxes which are numbered 1, 2 and 3. There are two more boxes but both of them are numbered 4. In how many different ways can we place the five balls in the given five boxes such that each box contains exactly one ball?

Try like this:
First choose what goes in the two identical boxes - since they are identical, order doesn't matter, we just pick two of the five balls. The tow yellow balls are a complication though, so I'd count like this: 3 ways to pick non-yellow balls only (choose 2 out of 3), 3 ways to pick one yellow one non-yellow (choose 1 out of 3 - which is the non-yellow), and 1 way to pick 2 yellows: total of 7, but we need the breakdown details for the next part, so 3+3+1.
Then proceed to find in each of these types of configurations, in how many ways you can distribute the remaining 3 balls in the remaining 3 boxes.
