Question

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

• A. 3ⁿ
• B. 3ⁿ – 3
• C. 3ⁿ – 6
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Since each item can be dealt in 3 ways, therefore, the total number of ways = 3 x 3 x 3 ..n times = 3ⁿ.

Now, we find the number of ways in which either one or two boxes are empty.

When two boxes are empty, then all the items are put into the third box and hence there are three such ways.

When one box is empty, then all the items are put into the remaining two boxes neither of which is empty and this can be done in 2ⁿ – 2 ways. So only one box can be empty in 3(2ⁿ-2) ways.

Hence, the required number of ways

= 3ⁿ – 3 – 3 (2ⁿ – 2)

= 3ⁿ – 3 x 2ⁿ + 3