Question 939054: Why do we divide (n-1)! by 2 in key ring problems in circular permutation?
Answer by mathmate(429)   (Show Source):
You can put this solution on YOUR website!
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.
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