Eighteen guest have to be sated. Half on each side of long table. Four particlular guest desire to sit on one particular side and three others on the other side. Determine the number of ways in which seating arrangements can be made.

Answer

To solve the problem of seating 18 guests at a long table with specific seating preferences, we can break it down step by step.

### Step 1: Understand the seating arrangement
We have 18 guests in total, and they need to be seated on two sides of a long table, with 9 guests on each side.

### Step 2: Identify the guests with specific seating preferences
According to the problem:
- 4 particular guests want to sit on one side (let's call this Side A).
- 3 other particular guests want to sit on the other side (let's call this Side B).

This means:
- Side A will have 4 specific guests plus 5 more guests to make a total of 9.
- Side B will have 3 specific guests plus 6 more guests to make a total of 9.

### Step 3: Calculate the remaining guests
After seating the 4 guests on Side A and the 3 guests on Side B, we have:
- Total guests = 18
- Guests seated = 4 (Side A) + 3 (Side B) = 7
- Remaining guests = 18 - 7 = 11 guests.

### Step 4: Choose additional guests for each side
We need to select 5 more guests to sit on Side A from the remaining 11 guests. The number of ways to choose 5 guests from 11 is given by the combination formula \( \binom{n}{r} \), which represents the number of ways to choose \( r \) elements from a set of \( n \) elements.

So, we calculate:
\[ \text{Ways to choose 5 guests for Side A} = \binom{11}{5} \]

### Step 5: Calculate the arrangements for each side
After selecting the guests, we need to arrange them on each side:
- Side A will have 9 guests (4 specific + 5 chosen), which can be arranged in \( 9! \) ways.
- Side B will also have 9 guests (3 specific + 6 chosen), which can be arranged in \( 9! \) ways.

### Step 6: Combine the results
The total number of seating arrangements is the product of the ways to choose the guests for Side A and the arrangements for both sides:
\[ \text{Total arrangements} = \binom{11}{5} \times 9! \times 9! \]

### Final Calculation
Now, we can compute the values:
1. Calculate \( \binom{11}{5} \):
\[ \binom{11}{5} = \frac{11!}{5!(11-5)!} = \frac{11!}{5!6!} = 462 \]

2. Calculate \( 9! \):
\[ 9! = 362880 \]

3. Therefore, the total arrangements are:
\[ \text{Total arrangements} = 462 \times 362880 \times 362880 \]

### Conclusion
The total number of ways to arrange the guests is:
\[ \text{Total arrangements} = 462 \times (362880)^2 \]