A family consists of a grandfather, m sons and daughters and 2n grandchildren. They are to be seated in a row for dinner. The grand-children wish to occupy the n seats at each end and the grandfather refuses to have grand-children on either side of him. In how many ways can the family be made to sit.

Options

A. `(2n)!m!(m-1)`
B. `(2n)!m!m`
C. `(2n)!(m-1)!(m-1)`
D. `(2n-1)!m !(m-1)`

Answer

To solve the problem, we need to determine the number of ways to arrange a family consisting of a grandfather, m sons and daughters, and 2n grandchildren under specific seating conditions. Let's break down the solution step by step.

### Step 1: Arranging the Grandchildren
The grandchildren wish to occupy n seats at each end of the row. Since there are 2n grandchildren, we will fill these 2n seats with them.

- The total number of ways to arrange 2n grandchildren in these 2n seats is given by the factorial of the number of grandchildren:
\[ \text{Ways to arrange grandchildren} = (2n)! \]

### Step 2: Arranging the Sons and Daughters
After the grandchildren are seated, we need to arrange the m sons and daughters. The seating arrangement must ensure that the grandfather is not sitting next to any grandchildren.

- The m sons and daughters will occupy the seats between the grandchildren. Since there are n grandchildren on each end, the arrangement of grandchildren creates m gaps between them where the sons and daughters can sit.
- The total number of ways to arrange m sons and daughters in these m seats is:
\[ \text{Ways to arrange sons and daughters} = m! \]

### Step 3: Arranging the Grandfather
The grandfather cannot sit next to any grandchildren. Since the m sons and daughters are seated in the gaps created by the grandchildren, the grandfather can only sit in the gaps between the sons and daughters.

- There are m sons and daughters, which creates m - 1 gaps (one gap before the first son/daughter, one gap between each pair of sons/daughters, and one gap after the last son/daughter).
- Therefore, the grandfather can occupy any of these m - 1 gaps.

### Step 4: Total Arrangements
Now, we can combine all the arrangements:

- The total number of ways to arrange the grandchildren, sons and daughters, and the grandfather is:
\[ \text{Total arrangements} = (2n)! \times m! \times (m - 1) \]

### Final Answer
Thus, the total number of ways the family can be seated is:
\[ \text{Total arrangements} = (2n)! \times m! \times (m - 1) \]