For a party 7 guests are invited by a husband and his wife. They sit in a row for dinner. The probability that the husband and his wife sit together, is

Options

A. `(2)/(7)`
B. `(2)/(9)`
C. `(1)/(9)`
D. `(4)/(9)`

Explanation

To find the probability that the husband and wife sit together among 9 members (7 guests + 1 husband + 1 wife), we can follow these steps:

### Step 1: Calculate the total arrangements of 9 members
The total number of ways to arrange 9 members in a row is given by the factorial of 9:
\[ \text{Total arrangements} = 9! \]

### Step 2: Treat the husband and wife as a single unit
To find the arrangements where the husband and wife sit together, we can consider them as a single unit or block. This means we now have 8 units to arrange: the husband-wife block and the 7 guests.

### Step 3: Calculate the arrangements of the 8 units
The number of ways to arrange these 8 units is given by the factorial of 8:
\[ \text{Arrangements of 8 units} = 8! \]

### Step 4: Account for the internal arrangement of the husband and wife
Within their block, the husband and wife can sit in two different ways: either the husband is on the left and the wife on the right, or vice versa. Therefore, we multiply the arrangements of the 8 units by 2:
\[ \text{Arrangements with husband and wife together} = 2 \times 8! \]

### Step 5: Calculate the probability
The probability that the husband and wife sit together is the ratio of the arrangements where they sit together to the total arrangements:
\[ \text{Probability} = \frac{\text{Arrangements with husband and wife together}}{\text{Total arrangements}} = \frac{2 \times 8!}{9!} \]

### Step 6: Simplify the probability
We know that \(9! = 9 \times 8!\), so we can simplify the probability:
\[ \text{Probability} = \frac{2 \times 8!}{9 \times 8!} = \frac{2}{9} \]

Thus, the probability that the husband and wife sit together is:
\[ \frac{2}{9} \]

### Final Answer:
The probability that the husband and wife sit together is \(\frac{2}{9}\).

---