When arranging people in a circular table, the problem can be approached by fixing one person in place to break the circle and then arranging the remaining people around them. This is because, in a circle, rotations of the same arrangement are considered identical.
Here's the detailed step-by-step solution:
1. Understand the problem: We need to arrange 10 people around a circular table where rotations of the same arrangement are considered the same.
2. Conceptual approach:
- If it was a linear arrangement (a straight line), there would be [tex]\(10!\)[/tex] (10 factorial) ways to arrange 10 people.
- However, in a circular arrangement, fixing one person and then arranging the other 9 people is necessary to count each unique arrangement only once.
3. Fix one person's position: By fixing one person's position, the problem reduces to arranging the remaining 9 people in the remaining 9 seats.
4. Calculate the number of ways:
- The number of ways to arrange 9 people is [tex]\(9!\)[/tex] (9 factorial).
5. Select the correct answer:
- The correct formula for the number of arrangements in a circular table with 10 people is [tex]\((10-1)!\)[/tex].
Therefore, the answer is:
A. [tex]\((10-1)!\)[/tex]
