6 people abcdef are sitting at a round table  evenly spaced apart. How many ways are there for A and B to sit directly opposite each other 

Since the table is round, one person’s position can be fixed to account for the symmetry of the circle. Let’s fix person A in a particular seat. Now, there are only 5 remaining positions for the others.

To ensure that A and B sit directly opposite each other, we need to place B in the seat directly opposite A. Since A is fixed, there is exactly **1 way** to place B in the seat directly opposite A.

After placing A and B, there are 4 remaining people (C, D, E, F) to arrange in the remaining 4 seats. These 4 people can be arranged in \(4! = 24\) different ways.

Thus, the total number of ways to seat A and B directly opposite each other is:
\[
1 \times 4! = 24
\]

So, there are **24 ways** for A and B to sit directly opposite each other.

Would you like further details or have any other questions?

Here are 5 related questions to expand on this problem:

1. How many ways are there to seat all 6 people in any arrangement at a round table?
2. What if C and D also need to sit directly opposite each other? How many seating arrangements are possible then?
3. What if A, B, C, and D need to sit alternating around the table? How many ways can this be done?
4. How does the seating arrangement change if the table is not round but linear?
5. How many ways are there for A and B to sit next to each other at a round table?

**Tip**: For circular arrangements, fixing one person's position simplifies calculations by removing rotational symmetry.

6 people abcdef are sitting at a round table evenly spaced apart. How many ways are there for A and B to sit directly opposite each other?

Explore how to calculate the number of ways for A and B to sit directly opposite each other at a circular table with 6 people. Learn about combinatorics, circular permutations, and factorials with a step-by-step solution. Suitable for students in Grades 10-12.

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