there are 5 different caps `c_1 ,c_2 ,c_3 ,c_4 and c_5` and 5 different boxes ` B_1 ,B_2 ,B_3 ,B_4 and B_5 ` the capacity of each box is sufficient to accommodate all the the 5 caps .
If any box can have any number of caps, in how many ways can all the caps be distributed?

Options

A. 3125
B. 1235
C. 2358
D. 1248

Answer

To solve the problem of distributing 5 different caps \( C_1, C_2, C_3, C_4, C_5 \) into 5 different boxes \( B_1, B_2, B_3, B_4, B_5 \) where each box can hold any number of caps, we can follow these steps:

### Step 1: Understand the Distribution
Each cap can be placed in any of the 5 boxes. This means that for each cap, there are 5 choices.

### Step 2: Calculate Choices for Each Cap
- For cap \( C_1 \), there are 5 choices (it can go into any of the 5 boxes).
- For cap \( C_2 \), there are also 5 choices.
- This is the same for caps \( C_3, C_4, \) and \( C_5 \).

### Step 3: Multiply the Choices
Since the choices for each cap are independent, we can multiply the number of choices for each cap:
- Total choices = Choices for \( C_1 \) × Choices for \( C_2 \) × Choices for \( C_3 \) × Choices for \( C_4 \) × Choices for \( C_5 \)

This gives us:
\[ \text{Total choices} = 5 \times 5 \times 5 \times 5 \times 5 = 5^5 \]

### Step 4: Calculate \( 5^5 \)
Now, we need to calculate \( 5^5 \):
\[ 5^5 = 3125 \]

### Conclusion
Thus, the total number of ways to distribute the 5 different caps into the 5 different boxes is \( 3125 \).