Four persons A, B, C and D are to the seated at a circular table. In how many ways can they be seated ?

Answer

To find the number of ways to seat four persons A, B, C, and D at a circular table, we can follow these steps:

### Step 1: Understand Circular Permutations
In circular permutations, one position is fixed to avoid counting rotations of the same arrangement multiple times. This is because in a circle, rotating the arrangement does not create a new unique arrangement.

### Step 2: Fix One Person
We can fix one person in one position. Let's fix person A at one seat. This means we only need to arrange the remaining three persons (B, C, and D).

### Step 3: Arrange the Remaining Persons
Now, we need to arrange the remaining three persons (B, C, and D) around the table. The number of ways to arrange n distinct objects in a line is given by n!. However, since we have fixed one person, we will arrange the remaining (n-1) persons.

### Step 4: Calculate the Factorial
For our case, we have:
- Total persons = 4 (A, B, C, D)
- Fixed person = 1 (A)
- Remaining persons = 4 - 1 = 3 (B, C, D)

The number of arrangements of the remaining 3 persons is given by:
\[ (n-1)! = 3! \]

Calculating 3!:
\[ 3! = 3 \times 2 \times 1 = 6 \]

### Final Answer
Thus, the total number of ways to seat the four persons A, B, C, and D at a circular table is **6**.

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