Question

The number of ways in which 6 distinct things can be distributed into 2 boxes so that no box is empty is:

• A. 36
• B. 64
• C. 62
• D. 34

Hint:

When distributing items into boxes with no box left empty, calculate the total distributions and subtract the scenarios where any box remains empty. This ensures each box contains at least one item.

Answer 1

The Correct Option is C

Each of the 6 distinct items can be placed in one of the 2 boxes, giving a total of \(2^6 = 64\) ways to distribute all items. However, this total includes the cases where one of the boxes is empty. We need to subtract these cases to ensure that both boxes contain at least one item. There are 2 scenarios where one box is empty: all items are in box 1 or all items are in box 2. Each scenario is counted once, so we subtract 2 from 64: \[ 64 - 2 = 62 \] Thus, there are 62 ways to distribute the 6 items into 2 boxes such that no box is empty.