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 with the condition that two particular persons may not sit together, we can follow these steps:

### Step 1: Calculate the total arrangements of 7 persons at a round table.
In circular permutations, the number of ways to arrange \( n \) persons is given by \( (n - 1)! \).

For 7 persons:
\[ \text{Total arrangements} = (7 - 1)! = 6! = 720 \]

### Step 2: Calculate the arrangements where the two particular persons sit together.
Let's denote the two particular persons as \( P_1 \) and \( P_2 \). If we consider \( P_1 \) and \( P_2 \) as a single unit or block, we can treat them as one person. Therefore, we now have 6 units to arrange: \( (P_1P_2), P_3, P_4, P_5, P_6, P_7 \).

The number of ways to arrange these 6 units in a circular manner is:
\[ \text{Arrangements with } P_1 \text{ and } P_2 \text{ together} = (6 - 1)! = 5! = 120 \]

Since \( P_1 \) and \( P_2 \) can be arranged among themselves in \( 2! \) ways, we multiply the arrangements by \( 2! \):
\[ \text{Total arrangements with } P_1 \text{ and } P_2 \text{ together} = 5! \times 2! = 120 \times 2 = 240 \]

### Step 3: Subtract the arrangements where \( P_1 \) and \( P_2 \) sit together from the total arrangements.
Now, we subtract the number of arrangements where \( P_1 \) and \( P_2 \) are together from the total arrangements:
\[ \text{Arrangements where } P_1 \text{ and } P_2 \text{ do not sit together} = 720 - 240 = 480 \]

### Final Answer
Thus, the number of ways in which seven persons can be arranged at a round table if two particular persons may not sit together is:
\[ \boxed{480} \]