How many ways are there to put 5 balls in 2 boxes if the balls are not distinguishable but the boxes are?
This problem is about counting the number of partitions of 5 balls into 2 boxes, where the order of the boxes matters. One way to approach this problem is to think of the first box as having a variable number of balls from 0 to 5, and then the second box has the remaining balls. For example, if the first box has 0 balls, then the second box has 5 balls. If the first box has 1 ball, then the second box has 4 balls, and so on. This way, we can enumerate all the possible partitions by increasing the number of balls in the first box by one each time, until we reach 5 balls. The partitions are: (0,5), (1,4), (2,3), (3,2), (4,1), and (5,0). There are 6 partitions in total, so there are 6 ways to put 5 balls in 2 boxes if the balls are not distinguishable but the boxes are. 