There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. in how many ways can the guests be accommodated?

Answer

To solve the problem of accommodating four guests in eight rooms such that no two guests are in adjacent or opposite rooms, we can follow these steps:

### Step 1: Understand the Room Arrangement
We have 8 rooms arranged as follows:
- Rooms 1, 2, 3, 4 on one side of the corridor
- Rooms 5, 6, 7, 8 directly opposite to them on the other side

### Step 2: Identify Non-Adjacent and Non-Opposite Rooms
Since no two guests can be in adjacent rooms or in opposite rooms, we can visualize the arrangement:
- If a guest is in Room 1, they cannot be in Rooms 2, 5, or 6.
- If a guest is in Room 2, they cannot be in Rooms 1, 3, 5, or 6.
- If a guest is in Room 3, they cannot be in Rooms 2, 4, 7, or 8.
- If a guest is in Room 4, they cannot be in Rooms 3, 8, or 7.
- Similarly for Rooms 5, 6, 7, and 8.

### Step 3: Choose a Valid Arrangement
To accommodate the guests, we can choose one of two valid arrangements:
1. Rooms 1, 3, 5, 7
2. Rooms 2, 4, 6, 8

### Step 4: Calculate the Number of Ways to Arrange Guests
For each arrangement, we can place the guests in any of the chosen rooms. The number of ways to arrange 4 guests in 4 rooms is given by the factorial of the number of guests:
- The number of arrangements for each configuration is \(4!\).

### Step 5: Combine the Arrangements
Since there are two valid configurations (1, 3, 5, 7 and 2, 4, 6, 8), we multiply the number of arrangements by 2:
- Total arrangements = \(2 \times 4!\)

### Step 6: Calculate the Final Answer
Calculating \(4!\):
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
Thus, the total number of ways to accommodate the guests is:
\[ 2 \times 24 = 48 \]

### Final Answer
The total number of ways to accommodate the guests is **48**.

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