Question

In how many ways can $5$ balls be placed in $4$ boxes if the balls are distinguishable, and the boxes are indistinguishable?

Answer 1

If the balls are distinguishable and the boxes are indistinguishable, we need to count the number of ways to distribute the balls among the boxes.

We can use the concept of stars and bars to solve this problem. Imagine that we have 5 stars (representing the balls) and 3 bars (representing the divisions between the boxes). We need to place the bars to separate the stars into 4 groups, each representing a box.

For example, if we have the arrangement: |*||, it represents placing 2 balls in the first box, 1 ball in the second box, 0 balls in the third box, and 2 balls in the fourth box.

In this case, we have a total of 5 stars (balls) and 3 bars (divisions between boxes), so we have 8 objects in total. The number of ways to arrange these objects can be calculated using permutations.

The number of ways to arrange 8 objects (5 stars and 3 bars) is given by: 8! / (5! * 3!)

Simplifying this expression, we get: 8! = 8 * 7 * 6 * 5! 5! = 5 * 4 * 3 * 2 * 1

Canceling out the common factors, we have: 8! / (5! * 3!) = (8 * 7 * 6) / (3 * 2 * 1) = 56

Therefore, there are 56 ways to place 5 distinguishable balls in 4 indistinguishable boxes.