Six persons A, B, C, D, E and F are to be seated at a circular table . The number of ways this can be done if A must have either B or C on his right and B must have either C or D on his right is

Options

A. 36
B. 12
C. 24
D. 18

Answer

To solve the problem of seating six persons A, B, C, D, E, and F at a circular table with specific conditions, we can follow these steps:

### Step 1: Understand the Conditions
We have two conditions:
1. Person A must have either B or C on his right.
2. Person B must have either C or D on his right.

### Step 2: Identify Cases Based on Conditions
We can break the problem into cases based on who sits next to A and B:
- **Case 1**: A is next to B (A, B)
- **Case 2**: A is next to C (A, C)
- **Case 3**: B is next to C (B, C)
- **Case 4**: B is next to D (B, D)

### Step 3: Analyze Each Case
We will analyze the arrangements based on the above cases:

#### Case 1: A is next to B
- A can be seated as (A, B) or (B, A).
- If A is next to B, then B can have C or D on his right.
- If B has C on his right, the arrangement is (A, B, C).
- If B has D on his right, the arrangement is (A, B, D).

In both sub-cases, we have 3 remaining persons (C, D, E, F) to arrange around the table.

#### Case 2: A is next to C
- A can be seated as (A, C) or (C, A).
- If A is next to C, then B can have C or D on his right.
- If B has C on his right, the arrangement is (A, C, B).
- If B has D on his right, the arrangement is (A, C, D).

Again, we have 3 remaining persons (B, D, E, F) to arrange around the table.

### Step 4: Calculate Arrangements for Each Case
For each arrangement of A and B, we can arrange the remaining four persons in a circular manner. The formula for arranging n persons in a circular table is (n-1)!.

1. **For each of the cases (A, B) and (A, C)**:
- We have 4 persons left to arrange.
- The number of arrangements = (4-1)! = 3! = 6.

### Step 5: Total Arrangements
Now, we combine the arrangements from all cases:
- Case 1 (A next to B): 6 arrangements
- Case 2 (A next to C): 6 arrangements

Thus, the total number of arrangements = 6 + 6 + 6 = 18.

### Final Answer
The total number of ways to seat the six persons at a circular table under the given conditions is **18**.

---