How many ways are there to put 5 balls in 3 boxes if the balls are distinguishable but the boxes are not?
You want to distribute your 5 distinguishable balls into 3 indistinguishable boxes. Let \(B(5,3)\)denote the number of ways in which this can be done into exactly 3 indistinguishable non-empty boxes, and use the recurrence relation \(B(n,k)=B(n−1,k−1)+kB(n−1,k)\)with \(B(n,1)=1\) and \(B(n,n)=1\). You seek \(B(5,1)+B(5,2)+B(5,3)\).
This has something to do with "Stirling numbers" although I am not too sure on this topic. You might consider searching it up!