The number of ways to distribute 5 indistinguishable objects into 3 indistinguishable boxes is essentially the number of ways to find solutions to $ x_1+x_2+x_3 = 5 $, where $ x_1 $, $ x_2 $, and $ x_3 $ are non-negative integers. Since the boxes are indistinguishable, solution (a,b,c) is no different than (b,a,c), (c,a,b), etc. Therefore, our question is really how many ways we can split five into three parts, order doesn't matter. The possible distributions are (0,0,5), (0,2,3), (0,1,4), (1,2,2), and (1,1,3). Thus, there are five possible ways.
