What are the total number of ways in which $n$ distinct objects can be put into two different boxes so that no box remains empty? I have came across this problem which is my textbook.According to the book the answer is: $2^n - 2$. But i don't understand how they got to that answer. Can someone help me out?

There are $2^n$ different subsets that can be taken from a set of $n$ objects.  Pick any subset and put it in the first box.  Then put the rest in the other box.  That makes $2^n$ ways to put $n$ object in $2$ boxes.  But you have the restriction that neither box can be empty.  So you can't choose the empty set for box 1.
And you can't put the whole set of $n$ objects in box 1, because that leaves box 2 empty.  So there are $2^n-2$ ways that meet your criteria.
