Question

How many ways are there to put 6 balls in 3 boxes if the balls are distinguishable but the boxes are not?

Answer 1

Would you believe me if I said
\[\huge\sum_{i=0}^32^{6-i}\binom 3i=216~?\]

Answer 2

Hmm that might not be right. I think it considers the boxes to be distinguishable... Ugh

Answer 3

\(\dbinom3i\) counts the numbers of ways you can assign \(i=0,1,...,5,6\) balls in the first box, and \(\large 2^{6-i}\) counts the ways the other two boxes can hold the remaining \(6-i\) balls. I think the formula double counts in some places... I'll have to look this over later.