Question

The number of ways in which n distinct objects can be put into two different boxes so that no box remains empty is

• A. $2^n-1$
• B. $n^2-1$
• C. $2^n-2$
• D. $n^2-2$

Answer 1

The correct option is C $2^n-2$

Each object can be put either in box $B_1$ (say) or in box $B_2$(say). So, there are two choices for each of the n objects. Therefore the number of choices for n distinct objects is
$2\times 2\times \underset{n-times}{\cdots }\times 2=2^n$
One of these choices corresponds to either the first or the second box being empty.
Thus, there are $2^n-2$ ways in which neither box is empty.