PRMO 2019 question 5 - circular table

 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 ?

Solution:
Try to arrange it manually.
Let's first put one of 2,3,4 because each of them has 2 adjacent numbers which we need to avoid whereas for 1 and 5 there is only 1 number to avoid. Let's start with 2.
      2
Now adjacent numbers can be only 4,5
            2
     4           5
Now 3 can't go next to 4 so:
            2
     4           5
        1    3
If you flip 4,5 initially
            2
     5           4
        3    1

So effectively, there are only 2 solutions.
Each of them can be rotated 4 times.
So total 10 solutions.