To solve the problem, we need to find the number of ways to accommodate four guests in eight rooms such that no two guests are in adjacent rooms or in opposite rooms. Let's break down the solution step by step.
### Step 1: Visualize the Room Arrangement
We have eight rooms arranged as follows:
```
Room 1 Room 2 Room 3 Room 4
| | | |
-------------------------
| | | |
Room 5 Room 6 Room 7 Room 8
```
Rooms 1, 2, 3, and 4 are on one side of the corridor, and rooms 5, 6, 7, and 8 are directly opposite them.
### Step 2: Identify Valid Room Combinations
Since no two guests can be in adjacent rooms or in opposite rooms, we need to choose rooms carefully. The valid combinations of rooms can be categorized based on the selection of rooms on one side of the corridor.
### Step 3: Choose Rooms
We can choose rooms from either side of the corridor. Let's denote the rooms on one side as A (rooms 1-4) and the rooms on the other side as B (rooms 5-8).
1. If we select rooms from side A, we can choose:
- Rooms 1, 3 (not adjacent or opposite)
- Rooms 2, 4 (not adjacent or opposite)
2. If we select rooms from side B, we can choose:
- Rooms 5, 7 (not adjacent or opposite)
- Rooms 6, 8 (not adjacent or opposite)
### Step 4: Count the Combinations
For each side, we have 2 valid combinations:
- (1, 3) from side A and (5, 7) from side B
- (2, 4) from side A and (6, 8) from side B
Thus, we have 2 ways to select rooms from side A and 2 ways from side B. Therefore, the total combinations of selecting rooms are:
\[
2 \text{ (from A)} \times 2 \text{ (from B)} = 4 \text{ combinations}
\]
### Step 5: Arrange the Guests
Once we have selected the rooms, we can arrange the four guests in those rooms. The number of ways to arrange 4 guests in 4 rooms is given by \(4!\) (4 factorial):
\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]
### Step 6: Calculate Total Arrangements
The total number of ways to accommodate the guests is:
\[
N = \text{(number of room combinations)} \times \text{(ways to arrange guests)} = 4 \times 24 = 96
\]
### Step 7: Find \( \frac{N}{6} \)
Now, we need to find the value of \( \frac{N}{6} \):
\[
\frac{N}{6} = \frac{96}{6} = 16
\]
### Final Answer
Thus, the value of \( \frac{N}{6} \) is:
\[
\boxed{16}
\]
