Five persons are to be seated around a circular table they are wearing badges of consecutive numbers 1,2,3,4,5. In how many ways can they be seated such that no two consecutive numbered badge wearing people are seated next to each other ?
My attempt: I got only 2 cases
- 13524
- 14253
But the answer is 10.
Apparently, $13524$ and $35241$ are to be counted as different seating arrangements, if we are to get to a count of $10$. So it seems that we are to count differing starting points as different seating arrangements.
This means your two solutions each become five by rotating the starting point.
Why do I not think there are more seating arrangements? $3$ cannot be adjacent to $2$ or $4$. Therefore, $3$ and its neighbors must be $135$ or $531$. Since $2$ cannot be next to $1$ and $4$ cannot be next to $5$, we have $41352$ and $25314$. (In both cases, we create the allowed adjacency between $2$ and $4$.) These are the Question's two answers, but with rotated starting points.