Question

Eight people are to be seated in a circular table, but 2 people do not want to sit beside each other. How many seating arrangements are possible?

Answer 1

Assume person A is in seat 1. The total number of seating arrangements satisfying the problem will be 8 times the number of seating arrangements where A is in seat 1.

With A in 1 B can be in seats 3-7 and everyone else in any of the six remaining seats. So the number of arrangements where A is in 1 is 6*6!. Multiply this by the 8 places A could have sat and bingo!

Answer 2

isn't the formula for circular seating (n-1)!

Answer 3

Depending on the definition of "arrangement". If I have a circular conference room table, and one of the seats is closest to the door it may matter where I sit. In arranging people in an abstact circle, the placement may not matter, just the order.

Answer 4

seats 3-7 is only 5 seats

Answer 5

@Zarkon yes, that is right. 8*5*6! if placement matters 5*6! if only order matters.

Answer 6

I look at it as 7!-2*6!=5*6!

Answer 7

7! is because of the formula for circular seating right?

Answer 8

7! ways to arrange the 8 people...2*6! the number of ways to have the 2people next to each other...subtract the two to get the answer

Answer 9

ahh that's a manipulative way to do it..

Answer 10

why is 2*6! the way to have 2 people next to each other?

Answer 11

i understand 6! is because you consider the 2 people beside each other as one entity so it becomes (7-1)! and 2 is i assume because they can be interchanged?

Answer 12

Think of the two people as a unit (one fat person who can be flipped two ways).

Answer 13

so..that's a yes...?

Answer 14

yes...I guess I should read your posts ;)

Answer 15

thank you. that was also the solution i saw 7! - 2*6! and i wanted to see why so

Answer 16

ah..ic