Need help with counting

[covert wasp]

How many ways can six people sit around a circular table if two particular
people must not sit next to each other?

Im so confused. Can someone pls help!

[covert wasp]

• Total circular arrangements: (6−1)!=120
• Arrangements where A & B sit together:
o Treat them as a unit ⇒ 5 people ⇒ (5−1)!=24
o A&B can swap: ×2 ⇒ 24x 2= 48
So, required = 120 – 48 = 72

I got this answer from chatgpt

Not sure if this is correct. This doesnt make sense to me honestly

[mental elm]

it's pretty rough, but here's my though process on this.
you would first assign the one of two kids to not sit with each other in a seat. that's 6 spots already
then the other kid who can't sit only has 3 spots to chose from, because 2 seats on either side of the first kid cannot to be picked from.
and the the remaining 4 friends are free to sit where ever.
since they are being assigned a seat the order matters. So that would be a permutation

i'm a little rusty on the actual calculations but i believe after you find 4P4 you would just add.

I am not 100% on the calculating, but I believe the logic of the seating arrangement is correct

[lime willow]

I'm pretty sure this is correct

it makes sense to me logically