Seven persons are to be seated in a row. The probability that two particular persons sit next to each other is

Options

A. `(1)/(3)`
B. `(1)/(6)`
C. `(2)/(7)`
D. `(1)/(2)`

Explanation

To solve the problem of finding the probability that two particular persons sit next to each other when seven persons are seated in a row, we can follow these steps:

### Step 1: Define the total number of arrangements
First, we need to find the total number of ways to arrange 7 persons in a row. This is given by the factorial of the number of persons, which is:

\[ \text{Total arrangements} = 7! = 5040 \]

### Step 2: Group the two particular persons
Next, we consider the two particular persons (let's say A3 and A4) as a single unit or block since we want them to sit next to each other. By doing this, we reduce the problem to arranging 6 units: the block (A3 and A4) and the other 5 persons (A1, A2, A5, A6, A7).

### Step 3: Calculate the arrangements of the grouped persons
Now, we can arrange these 6 units. The number of arrangements of these 6 units is:

\[ \text{Arrangements of 6 units} = 6! = 720 \]

### Step 4: Arrange the two persons within the block
Within the block of A3 and A4, these two persons can be arranged in 2 different ways (A3 can be on the left or A4 can be on the left). Thus, we multiply the arrangements of the 6 units by the arrangements of A3 and A4:

\[ \text{Total arrangements with A3 and A4 together} = 6! \times 2! = 720 \times 2 = 1440 \]

### Step 5: Calculate the probability
Finally, we can find the probability that A3 and A4 sit next to each other by dividing the number of favorable arrangements by the total arrangements:

\[ \text{Probability} = \frac{\text{Number of favorable arrangements}}{\text{Total arrangements}} = \frac{1440}{5040} \]

### Step 6: Simplify the probability
Now we simplify the fraction:

\[ \frac{1440}{5040} = \frac{2}{7} \]

### Final Answer
Thus, the probability that the two particular persons sit next to each other is:

\[ \frac{2}{7} \]