How many ways are there to put 5 balls in 2 boxes if the balls are not distinguishable and neither are the boxes?
Since the balls and the boxes are not distinguishable, we only care about how many balls are in each box, not which balls or which box. Let x be the number of balls in one box and y be the number of balls in the other box. Then we have x + y = 5 and x, y are non-negative integers. We can solve this equation by using the stars and bars method, which says that the number of solutions is equal to the number of ways to arrange 5 stars (representing the balls) and 1 bar (representing the division between the boxes) in a line. For example, one possible arrangement is *|****, which means x = 1 and y = 4. The number of ways to arrange 6 symbols in a line is 6!, but we have to divide by 5! to account for the fact that the stars are not distinguishable, and by 2 to account for the fact that the boxes are not distinguishable. Therefore, the number of ways to put 5 balls in 2 boxes is 6! / (5! * 2) = 3. 