The number of ways in which n distinct balls can be put into three boxes, is

Options

A. 3n
B. `n^(3)`
C. `3^(n)`
D. `n+3`

Explanation

To solve the problem of determining the number of ways in which \( n \) distinct balls can be put into three boxes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have \( n \) distinct balls and 3 distinct boxes. Each ball can be placed in any of the three boxes.

2. **Choices for Each Ball**: For each ball, we have 3 choices (Box 1, Box 2, or Box 3).

3. **Calculating Total Choices**: Since the choices for each ball are independent of the choices for the other balls, we can multiply the number of choices for each ball.
- For the first ball, there are 3 choices.
- For the second ball, there are also 3 choices.
- This continues for all \( n \) balls.

4. **Using Exponentiation**: The total number of ways to place all \( n \) balls into the boxes can be expressed as:
\[ 3 \times 3 \times 3 \times \ldots \text{ (n times)} = 3^n \]

5. **Conclusion**: Therefore, the total number of ways to distribute \( n \) distinct balls into 3 boxes is \( 3^n \).

### Final Answer:
The number of ways in which \( n \) distinct balls can be put into three boxes is \( 3^n \).

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