How many ways are there to put 5 balls in 3 boxes if the balls are distinguishable but the boxes are not?
This is a classic example of a combinatorial problem, specifically a problem of distributing objects into groups. Since the boxes are indistinguishable, we don't need to worry about which box the balls go in; all that matters is how many balls are in each box.
We can approach this problem using the stars and bars method. Imagine that we have five balls (which are distinguishable) and two dividers (which are indistinguishable). We can arrange these seven objects in a line in any order. For example:
| * * | * * *
In this arrangement, the first box has 0 balls, the second box has 2 balls, and the third box has 3 balls. We can count the number of arrangements by counting the number of ways to choose 2 positions out of 7 for the dividers. Once the positions of the dividers are fixed, the balls will naturally fall into the boxes defined by the dividers.
The number of ways to choose 2 positions out of 7 is given by the binomial coefficient:
C(7, 2) = 7! / (2! * 5!) = 21
Therefore, there are 21 ways to put 5 balls in 3 indistinguishable boxes.