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 5 caps.
If all the caps are of different colors and each box can have only one cap, in how many ways can you arrange the caps among the 5 boxes?

Options

A. 120
B. 180
C. 360
D. none of these

Answer

To solve the problem of arranging 5 different caps into 5 different boxes, where each box can hold only one cap, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Elements**: We have 5 different caps: \( c_1, c_2, c_3, c_4, c_5 \) and 5 different boxes: \( B_1, B_2, B_3, B_4, B_5 \).

2. **Understand the Arrangement**: Each box can only contain one cap, and since we have exactly 5 caps and 5 boxes, each box will contain exactly one cap.

3. **Determine the Number of Arrangements**: The problem requires us to find the number of ways to arrange the 5 caps into the 5 boxes. This is a permutation problem because the order in which the caps are placed into the boxes matters.

4. **Calculate the Permutations**: The number of ways to arrange \( n \) distinct objects is given by \( n! \) (n factorial). In this case, \( n = 5 \).

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

5. **Conclusion**: Therefore, the total number of ways to arrange the 5 caps among the 5 boxes is \( 120 \).

### Final Answer:
The total number of arrangements is **120**.

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