The number of ways in $n$ distinct objects can be put into two identical boxes, so that neither box remains empty.
My Try:: If the question is the numbers of ways in $n$ distinct objects can be put into two Distinct boxes so that no box remains empty, then I can solve easily; this can be done in $2^n-2$ ways
But I do not Understand how can I solve the original Question.
Best Answer
The only change needed if the two boxes are identical is to divide the number of ways by two, since swapping boxes produces two different cases when the boxes (as well as the objects) are distinct.
So $2^{n-1} -1$ ways to fill the two identical boxes, subject to having neither empty.