A family consists of a grandfather, 5 sons and daughters and 8 grand child. They are to be seated in a row for dinner. The grand children wish to occupy the 4 seats at each end and the grandfather refuses to have a grandchild on either side of him. In how many ways can the family be made to sit?

Explanation

To solve the problem, we need to arrange a family consisting of a grandfather, 5 sons and daughters, and 8 grandchildren in a row for dinner, with specific seating restrictions. Let's break down the solution step by step.

### Step 1: Determine the total number of family members
The family consists of:
- 1 Grandfather
- 5 Sons and Daughters
- 8 Grandchildren

Total family members = 1 + 5 + 8 = 14

### Step 2: Identify seating restrictions
- The grandchildren wish to occupy the 4 seats at each end of the table.
- The grandfather refuses to have a grandchild on either side of him.

### Step 3: Arrange the grandchildren
Since the grandchildren occupy the 4 seats at each end, we have:
- 4 seats on the left end
- 4 seats on the right end

The total number of grandchildren is 8, and we need to choose 4 for the left end and 4 for the right end.

The number of ways to choose 4 grandchildren from 8 for the left end is given by the combination formula:
\[ \text{Ways to choose 4 from 8} = \binom{8}{4} \]

After choosing the grandchildren, we can arrange them in the 4 seats at each end. The number of ways to arrange 4 grandchildren in 4 seats is:
\[ 4! \text{ (for left end)} \times 4! \text{ (for right end)} \]

### Step 4: Arrange the grandfather and the sons and daughters
Now, we have 6 remaining seats in the middle (since 4 on the left and 4 on the right are occupied by grandchildren). The grandfather must sit in one of these 6 seats, and he cannot have a grandchild on either side of him.

This means that the grandfather can only sit between the sons and daughters. Since there are 5 sons and daughters, they create 4 gaps between them where the grandfather can sit.

The number of ways to choose a gap for the grandfather is:
\[ 4 \text{ (gaps)} \]

### Step 5: Arrange the sons and daughters
The remaining 5 sons and daughters can be arranged in the remaining 5 seats. The number of ways to arrange them is:
\[ 5! \]

### Step 6: Calculate the total arrangements
Now, we can combine all the arrangements:
1. Choose and arrange grandchildren:
\[ \binom{8}{4} \times 4! \times 4! \]
2. Choose a gap for the grandfather:
\[ 4 \]
3. Arrange the sons and daughters:
\[ 5! \]

Putting it all together, the total number of arrangements is:
\[ \text{Total arrangements} = \binom{8}{4} \times 4! \times 4! \times 4 \times 5! \]

### Final Calculation
Calculating each component:
- \(\binom{8}{4} = 70\)
- \(4! = 24\)
- \(5! = 120\)

Now substituting these values:
\[ \text{Total arrangements} = 70 \times 24 \times 24 \times 4 \times 120 \]

Calculating this gives:
\[ = 70 \times 576 \times 480 \]

Thus, the final answer is:
\[ \text{Total arrangements} = 480 \times 8! \]

### Conclusion
Hence, the required number of ways the family can be seated is:
\[ \text{Required number of ways} = 480 \times 8! \]