How many ways are there to put 4 balls in 3 boxes if two balls are indistinguishably green, two are indistinguishably red, and the boxes are distinguishable?
I need to count the number of different arrangements of balls and boxes, but I don't care about the order of the balls within a box, only the colors and the quantities. One way to approach this problem is to consider how many ways I can distribute the green balls among the boxes, and then multiply that by how many ways I can distribute the red balls among the boxes. To distribute the green balls, I can either put both in the same box, or split them into different boxes. If I put both in the same box, there are 3 choices for which box to use. If I split them into different boxes, there are 3 choices for the first box, and then 2 choices for the second box, but I have to divide by 2 to avoid counting the same pair twice. So, the total number of ways to distribute the green balls is 3 + 3 * 2 / 2 = 6. Similarly, the total number of ways to distribute the red balls is also 6. Therefore, the total number of ways to put 4 balls in 3 boxes is 6 * 6 = 36. 