How many ways are there to put $2$ white balls and $2$ black balls into $3$ boxes, given that balls of the same color are indistinguishable, but the boxes are distinguishable?

1. Let the three boxes be box 1, box 2, box 3

consider the 2 white balls, they can be placed in the 3 boxes as follows:

BOX1: BOX2: BOX3:

__2___ ___0__ __0___

__0___ ___2__ ___0__

__0___ ___0__ ___2__

___0__ ___1__ ___1__

__1___ ___0__ ___1__

__1___ ___1__ ___0__

so there are 6 possibilities to place the white balls. Similarly there are 6 ways to place the black balls. We can combine any position of the black balls with any position of the white balls, so there are in total 6*6=36 ways to place the balls.

Answer: 36