A meeting is held with eight people around a circular table. Two of the participants, April and May, want to sit so that there is exactly one other person between them. How many different seatings are possible? (Two seatings are considered the same if one can be rotated to obtain the other.)

SoggyPerson May 1, 2020

#1**+2 **

We seat April first. She has 8 choices. We seat May next. She has 2 choices: two spots to the left of April or two spots to the right of April. Then, the remaining six members can sit in ways. This appears to give a total of 8*2*6! seatings. However, because two seatings are considered the same if one is a rotation of the other, each distinct seating is counted 8 times in this product, once for each possible location of April. So, we must divide by 8 to give a total of **2*6! = 2*720 = 1440 seatings.**

We might have instead started by noting that it doesn't matter where April sits. She starts by sitting anywhere, and we count the number of ways the members sit relative to April. May still has 2 choices: two spots to the left of April or two to the right of April. The remaining six participants then can be seated in 6! ways, for a total of 2*6! = 1440 seatings, as before.

LuckyDucky May 1, 2020

#1**+2 **

Best Answer

We seat April first. She has 8 choices. We seat May next. She has 2 choices: two spots to the left of April or two spots to the right of April. Then, the remaining six members can sit in ways. This appears to give a total of 8*2*6! seatings. However, because two seatings are considered the same if one is a rotation of the other, each distinct seating is counted 8 times in this product, once for each possible location of April. So, we must divide by 8 to give a total of **2*6! = 2*720 = 1440 seatings.**

We might have instead started by noting that it doesn't matter where April sits. She starts by sitting anywhere, and we count the number of ways the members sit relative to April. May still has 2 choices: two spots to the left of April or two to the right of April. The remaining six participants then can be seated in 6! ways, for a total of 2*6! = 1440 seatings, as before.

LuckyDucky May 1, 2020

#2**+2 **

Just "anchor" April and May in any two seats with one seat in between

Note that they can be seated in two ways

The other people can be seated in 6! ways

So.....the total arrangements = 2 * 6! = 2 * 720 = 1440

CPhill May 1, 2020

#4**+2 **

Omg I'm so sorry your my idol in this website btw so I try to be more like you

so sorry

LuckyDucky
May 1, 2020

#5**+2 **

Nah you have the most thunder in this website while I just got on the leaderboard :D

LuckyDucky
May 1, 2020