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.

36