Question 939054
  
In a circular table problem, n people sit around the table in (n-1)! ways.
A sitting on the left of B counts as different from A sitting on the right of B.
  
The key-ring problem is similar to the circular table problem, except that the key ring can be reversed any time we want, so the left/right does not count any more.  Therefore the number of permutations has to be divided by two, giving (n-1)!/2.