Question

10 people are sitting in a circle. How many different ways can they be seated relative to each other if Steve and Alice must be seated directly across from each other?

Do I need to apply the circular permutation formula of (n-1)!, and do I need to multiply the end result by 2 to account for the fact that they could be in different positions kinda like standing beside each other where one is on the left but could also be on the right?

Answer 1

To solve this problem, we will think in the same way from where the formula of circular permutation came from. For circular permultation.. we assume 1 person fixed, then calculate no of ways other persons have choice to seat. We will think same in this case also. But in this case we will fix 2 persons(Steve and Alice) across each other.

So no of remaining people= 10-2= 8.

Now, Think line linear permutaion among those 8 persons.

So, 1st person has 8 choices to seat.

2nd person has 7 choices to seat.

3rd person has 6 choises to seat.

..

..

..

8 person have 1 choice to seat

So, total no of ways = 8*7*6*5*4*3*2*1= 8!, which is actually linear permutaion among those 8 people.

Now, Finally If we consider Steve and Alice, if they shuffle themselves, the case is actually rotating every case 180 degree about the circle. Rotating about the circle doesnot change the arrangement. So the total no of ways doesnot increase if we shuffle Steve and Alice.

So, The total no of ways which we require is 8!.

So, the required no of ways is 8!.

Please see the attached image to understand visually.