In how many ways can 8 students be seated around a round table?

When seating people around a round table, we usually account for rotational symmetry, meaning that rotating the entire arrangement doesn’t create a new seating arrangement.

For \(n\) people seated around a round table, the number of distinct seating arrangements is:

\[
(n - 1)!
\]

This is because fixing one person (to eliminate rotational symmetry) leaves \(n-1\) people to arrange.

In this case, for 8 students, the number of ways they can be seated around the table is:

\[
(8 - 1)! = 7! = 5040
\]

So, there are **5040** ways to seat 8 students around a round table.

Would you like a deeper explanation on how this formula works or have any other questions?

Here are 5 related questions for further exploration:
1. How many ways can 10 students be seated around a round table?
2. What if the 8 students need to be seated in a straight line instead of a round table?
3. How many ways can the 8 students be seated around a round table if two specific students must sit next to each other?
4. How many distinct seating arrangements are there if the 8 students are split into two equal groups at two round tables?
5. How would the number of seating arrangements change if the table was rectangular instead?

**Tip**: When calculating permutations or combinations in circular arrangements, always consider whether rotational symmetry matters!

Discover how to calculate the number of distinct seating arrangements for 8 students around a round table using permutations. Learn the formula (n - 1)! and why rotational symmetry affects the number of possible configurations. Suitable for students in Grades 9-12.

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