First slide

Permutations

Question

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

Moderate

A

$2^{n} - 1$
B

$n^{2} - 1$
C

$2^{n} - 2$
D

$n^{2} - 2$

Solution

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 then objects.
Therefore, the number of choices for $n$ distinct objects is
$2 \times 2 \times \dots \times 2 = 2^{n}$
These choices correspond to either the first or the second box
being empty. Thus, there are $2^{n} - 2$ ways in which neither box is empty.

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