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counting and probability intermediate

Problem

How many ways are there to put 5 balls in 2 boxes if the balls are distinguishable but the boxes are not?
Solution
Since the boxes are indistinguishable, there are 3 possibilities for arrangements of the number of balls in each box.

Case 1: 5 balls in one box, 0 in the other box. We must choose 5 balls to go in one box, which can be done in way.

Case 2: 4 balls in one box, 1 in the other box. We must choose 4 balls to go in one box, which can be done in ways.

Case 3: 3 balls in one box, 2 in the other box. We must choose 3 balls to go in one box, which can be done in ways.

This gives us a total of arrangements.
Final answer
16