There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrical situated (that is each room is exactly opposite to one other room). Four guests have to be accommodation In four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. If N is the number of ways in which guests can be accommodated. Then the value of `N/6` is

Answer

To solve the problem of accommodating 4 guests in 8 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:
- Side A: Room 1, Room 2, Room 3, Room 4
- Side B: Room 5, Room 6, Room 7, Room 8

Rooms 1 and 5 are opposite, Rooms 2 and 6 are opposite, Rooms 3 and 7 are opposite, and Rooms 4 and 8 are opposite.

### Step 2: Identify Non-Adjacent and Non-Opposite Room Combinations
Since guests cannot be in adjacent or opposite rooms, we can analyze the possible configurations:
- If we choose a room from Side A, we cannot choose the room directly opposite it on Side B or the adjacent rooms on the same side.

### Step 3: Choose Rooms
We can choose rooms in two distinct configurations:
1. Choose rooms from one side only (either Side A or Side B).
2. Choose rooms from both sides while ensuring they are not adjacent or opposite.

#### Configuration 1: All from Side A
- Possible selections: (1, 3, 4) or (2, 4, 3) or (1, 2, 4)
- Valid combinations: {1, 3, 4} (3 guests)

#### Configuration 2: Mixed Selection
- Choose one room from Side A and two from Side B, ensuring they are not adjacent or opposite.
- Valid combinations: {1, 5, 6}, {1, 7, 8}, {2, 5, 6}, {2, 7, 8}, etc.

### Step 4: Count Valid Combinations
After analyzing the configurations, we find that there are only two valid arrangements:
1. Rooms 1, 3, 5, 7
2. Rooms 2, 4, 6, 8

### Step 5: Arrange Guests
For each valid arrangement of rooms, we can arrange the 4 guests in those rooms. The number of ways to arrange 4 guests in 4 rooms is given by \(4!\) (factorial of 4).

### Step 6: Calculate Total Arrangements
Total arrangements \(N\) can be calculated as:
\[ N = \text{Number of valid room arrangements} \times \text{Number of arrangements of guests} \]
\[ N = 2 \times 4! = 2 \times 24 = 48 \]

### Step 7: Calculate \(N/6\)
Now, we need to find \(N/6\):
\[ N/6 = 48/6 = 8 \]

### Final Answer
Thus, the value of \(N/6\) is \(8\).

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