In how many 8 people can sit around a circular table so that Abhishek and Ayush should never sit opposite to each other.

Options

A. 3600
B. 7200
C. 1440
D. 720

Answer

To solve the problem of how many ways 8 people can sit around a circular table such that Abhishek and Ayush never sit opposite to each other, we can follow these steps:

### Step 1: Calculate the total arrangements without restrictions
In a circular arrangement, the number of ways to arrange \( n \) people is given by \( (n-1)! \). For 8 people, the number of arrangements is:

\[ (8-1)! = 7! = 5040 \]

### Step 2: Calculate the arrangements where Abhishek and Ayush sit opposite each other
When Abhishek and Ayush sit opposite each other, we can treat them as a single unit or block. This means we now have 6 other people plus the block of Abhishek and Ayush, making a total of 7 units to arrange.

The number of ways to arrange these 7 units in a circle is:

\[ (7-1)! = 6! = 720 \]

Since Abhishek and Ayush can switch places within their block, we multiply by 2:

\[ 720 \times 2 = 1440 \]

### Step 3: Subtract the restricted arrangements from the total arrangements
Now, to find the number of arrangements where Abhishek and Ayush do not sit opposite each other, we subtract the arrangements where they do sit opposite from the total arrangements:

\[ 5040 - 1440 = 3600 \]

### Final Answer
Thus, the number of ways 8 people can sit around a circular table such that Abhishek and Ayush never sit opposite each other is:

\[ \boxed{3600} \]

---