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 `C_1` and `C_5` are similar in all aspects, in how many ways can you arrange the caps in such a way that all the boxes have one cap?

Options

A. A) 70
B. B) 60
C. C) 75
D. D) 80

Answer

To solve the problem, we need to determine the number of ways to arrange 5 caps (C1, C2, C3, C4, C5) into 5 boxes (B1, B2, B3, B4, B5) given that caps C1 and C5 are identical.

### Step-by-Step Solution:

1. **Identify the Total Caps and Boxes**:
We have 5 different caps: C1, C2, C3, C4, and C5. However, C1 and C5 are identical.

2. **Understanding the Arrangement**:
We need to place one cap in each of the 5 boxes. Since C1 and C5 are identical, we will treat them as one type of cap for the purpose of counting arrangements.

3. **Calculate the Total Arrangements**:
If all caps were distinct, the total arrangements would be calculated using the factorial of the number of caps (5!):
\[ 5! = 120 \]

4. **Adjust for Identical Caps**:
Since C1 and C5 are identical, we need to divide the total arrangements by the factorial of the number of identical items (2! for C1 and C5):
\[ \text{Total arrangements} = \frac{5!}{2!} \]

5. **Perform the Calculation**:
\[ \frac{5!}{2!} = \frac{120}{2} = 60 \]

6. **Conclusion**:
Therefore, the total number of ways to arrange the caps in the boxes is **60**.

### Final Answer:
The total number of ways to arrange the caps is **60**.

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