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.
In how many arrangements does `B_1` have cap `C_1`?

Options

A. 5!
B. `5^4`
C. `^5P_4`
D. none of these

Answer

To find the number of arrangements where cap \( C_1 \) is placed in box \( B_1 \), we can follow these steps:

### Step 1: Place Cap \( C_1 \) in Box \( B_1 \)
Since we are given that cap \( C_1 \) must be in box \( B_1 \), we start by placing it there. This step has only 1 way to be done since \( C_1 \) must go into \( B_1 \).

### Step 2: Determine Remaining Caps
After placing cap \( C_1 \) in box \( B_1 \), we have 4 caps left: \( C_2, C_3, C_4, \) and \( C_5 \).

### Step 3: Count the Options for Remaining Caps
Each of the remaining 4 caps can be placed in any of the 5 boxes (including \( B_1 \)).

- For cap \( C_2 \), there are 5 choices (it can go into any of the 5 boxes).
- For cap \( C_3 \), there are also 5 choices.
- For cap \( C_4 \), there are again 5 choices.
- For cap \( C_5 \), there are still 5 choices.

### Step 4: Calculate Total Arrangements
Since the placement of each cap is independent of the others, we multiply the number of choices for each cap:

\[ \text{Total arrangements} = 5 \times 5 \times 5 \times 5 = 5^4 \]

### Final Answer
Thus, the total number of arrangements where cap \( C_1 \) is in box \( B_1 \) is:

\[ 5^4 \]