Try this beautiful problem from the PRMO, 2019 based on Two arrangements.
Five persons wearing badges with numbers 1,2,3,4,5 are seated on 5 chairs around a circular table. Find the number of ways they be seated that no two persons whose badges have consecutive numbers are seated next to each other.
But try the problem first...
Answer: is 10.
PRMO, 2019, Question 5
Combinatorics by Brualdi
Here they may seat in circular way such that 1,3,5,2,4 and again 1
then they may seat in another circular way 1,4,2,5,3 and again 1
Then number of ways = (2)(5) =10.