### Instruction: In how many ways can 8 people sit around a round table if Pierre and Thomas want to sit together, but Rosa doesn't want to sit next to either of them? (Treat rotations as not distinct but reflections as distinct.).### Response: This is a counting problem, so I want to find a systematic way to count all the possible arrangements that satisfy the given conditions.
I will start by fixing Pierre and Thomas as a pair, and then think of how to place the other 6 people around the table.
Since the table is round, the pair can be placed anywhere without changing the arrangement, so I don't need to worry about their position.
However, I do need to consider their orientation, since a reflection would count as a different arrangement.
So there are 2 ways to orient the pair, either Pierre is on the left of Thomas or vice versa.
Now, how can I place the other 6 people?
Well, I have 6 spots left around the table, but I need to avoid placing Rosa next to the pair.
So I have to exclude the 2 spots that are adjacent to the pair, leaving me with 4 spots to choose from for Rosa.
Once I choose a spot for Rosa, I have 5 people and 5 spots left.
The order of the people matters, so I can use the factorial notation to count the number of ways to arrange them.
There are 5! ways to arrange 5 people in 5 spots.
So, putting everything together, I have 2 ways to orient the pair, 4 ways to choose a spot for Rosa, and 5! ways to arrange the rest of the people.
Multiplying these gives me the total number of ways to sit the 8 people around the table.
Therefore, the answer is: