The number of ways to distribute n ball in two boxes such that no box remains empty when

Question

Find the number of ways to distribute $n$ balls in two boxes such that no box remains empty when

1. both boxes are different
2. both are identical

ANSWER

1. I solved first part by using logic that each ball will have two options (Box 1, Box 2), and $n$ balls will have $2^n$ ways. But this will include 2 cases in which either box 1 or 2 is not filled. Therefore, it's answer will be $2^n - 2$.

2. What change will occur by identical boxes? According to me it's answer must be same as in (1), but in textbook the answer is $2^{n-1} - 1$.

Please explain me what changes will occur by using identical boxes!

Answer 1

The first case makes an ordering out of the boxes. In other words, your result is an ordered pair, i.e. cutting $7$ objects, $(3,4) \ne (4,3)$.

If you cannot distinguish the boxes, the result is a set, and $\{3,4\} = \{4,3\}$ so the first answer for this case is a massive overcount, but by how much?