If a Grand father along with his two grandsons and four grand daughters are to be seated in a line for a photograph so that he is always in the middle then the probability that his two grandsons are never adjacent to each other is

Options

A. `(6!-4!. 4 . 2!)/(6!)`
B. `192/720`
C. `528/720`
D. `4/15`

Answer

To solve the problem step by step, we will follow the outlined procedure to find the probability that the two grandsons are never adjacent to each other when seated in a line with their grandfather and four granddaughters.

### Step 1: Understand the Arrangement
We have a total of 7 individuals: 1 grandfather, 2 grandsons, and 4 granddaughters. The grandfather must always be in the middle of the arrangement.

### Step 2: Fix the Grandfather's Position
Since the grandfather is always in the middle, he occupies the 4th position in a line of 7. This leaves us with 6 positions to fill with the 2 grandsons and 4 granddaughters.

### Step 3: Total Arrangements Without Restrictions
The total number of ways to arrange the 6 remaining individuals (2 grandsons and 4 granddaughters) is given by the factorial of the number of individuals:
\[ 6! = 720 \]

### Step 4: Arrangements with Grandsons Together
To find the arrangements where the grandsons are together, we can treat the two grandsons as a single unit or block. This gives us:
- 1 block of grandsons
- 4 granddaughters

This results in 5 units to arrange (the grandson block + 4 granddaughters). The number of ways to arrange these 5 units is:
\[ 5! = 120 \]
Since the grandsons can be arranged within their block in \(2!\) ways, the total arrangements where the grandsons are together is:
\[ 5! \times 2! = 120 \times 2 = 240 \]

### Step 5: Arrangements with Grandsons Not Together
To find the arrangements where the grandsons are not together, we subtract the arrangements where they are together from the total arrangements:
\[ \text{Arrangements where grandsons are not together} = 6! - (5! \times 2!) \]
Substituting the values we calculated:
\[ = 720 - 240 = 480 \]

### Step 6: Calculate the Probability
The probability that the two grandsons are never adjacent to each other is given by the ratio of the favorable outcomes to the total outcomes:
\[ P(\text{grandsons not together}) = \frac{\text{Arrangements where grandsons are not together}}{\text{Total arrangements}} = \frac{480}{720} = \frac{2}{3} \]

### Final Answer
Thus, the probability that the two grandsons are never adjacent to each other is:
\[ \frac{2}{3} \]