Find the total number of ways in which n distinct objects can be put into two different boxes so that no box remains empty.

Options

A. `2^(n)`
B. `2^(n)-1`
C. `2^(n)-2`
D. `2^(n)-3`

Explanation

To find the total number of ways in which \( n \) distinct objects can be put into two different boxes such that no box remains empty, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have \( n \) distinct objects and 2 distinct boxes. We want to distribute these objects into the boxes in such a way that neither box is empty.

2. **Calculating Total Distributions**: Each object can go into either of the two boxes. Therefore, for each of the \( n \) objects, there are 2 choices (Box 1 or Box 2). This gives us a total of:
\[ 2^n \]
ways to distribute \( n \) objects into 2 boxes.

3. **Excluding Empty Box Cases**: However, the problem states that no box can remain empty. The total distribution we calculated includes cases where all objects might be in one box (leaving the other box empty). There are exactly 2 such cases:
- All \( n \) objects in Box 1 (Box 2 is empty).
- All \( n \) objects in Box 2 (Box 1 is empty).

4. **Final Calculation**: To find the valid distributions where no box is empty, we need to subtract these 2 cases from the total distributions:
\[ \text{Valid distributions} = 2^n - 2 \]

5. **Conclusion**: Therefore, the total number of ways to distribute \( n \) distinct objects into 2 different boxes such that no box remains empty is:
\[ 2^n - 2 \]