The number of ways in which seven persons can be arranged at a round table if two particular persons may not sit together, is

Options

A. 480
B. 120
C. 80
D. none of these

Answer

To solve the problem of arranging seven persons at a round table such that two particular persons do not sit together, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate Total Arrangements without Restrictions**:
When arranging \( n \) persons at a round table, the formula used is \( (n-1)! \). For 7 persons:
\[ \text{Total arrangements} = (7-1)! = 6! = 720 \]

2. **Calculate Arrangements with Two Particular Persons Together**:
To find the arrangements where the two particular persons (let's call them A and B) sit together, we can treat A and B as a single entity or block. Thus, we now have 6 entities to arrange (the block AB and the other 5 persons).
\[ \text{Arrangements with A and B together} = (6-1)! = 5! = 120 \]
Since A and B can be arranged within their block in 2 ways (AB or BA), we multiply by 2:
\[ \text{Total arrangements with A and B together} = 5! \times 2 = 120 \times 2 = 240 \]

3. **Calculate Arrangements where A and B do not sit together**:
To find the arrangements where A and B do not sit together, we subtract the arrangements where they are together from the total arrangements:
\[ \text{Arrangements where A and B do not sit together} = \text{Total arrangements} - \text{Arrangements with A and B together} \]
\[ = 720 - 240 = 480 \]

### Final Answer:
The number of ways in which seven persons can be arranged at a round table if two particular persons may not sit together is **480**.