DarkKnight wrote:Why can't we apply the principle of arranging five people around a circular table?? According to that the answer should be (n-1)! = 4!= 24.
The number of ways to arrange
n elements around a circular table is (
n-1)!.
But the number of ways to arrange
n elements around a
ring is (
n-1)!/2.
The ring gives us 1/2 the number of distinct arrangements because, unlike a table,
a ring can be flipped over.
For example:
Given the 4 elements ABCD, there are (4-1)! = 3! = 6 ways to arrange them around a circular table.
Among these arrangements are ABCD and DBCA.
But around a ring, ABCD and DBCA represent the same arrangement, because when the ring is flipped over, ABCD becomes DBCA and DBCA becomes ABCD.
So around a ring, the number of distinct arrangements is cut in 1/2, as given in the formula above.
Assuming that -- like a ring -- the pentagon in the problem above can be flipped over, the number of distinct ways to arrange the 5 sides = (5-1)!/2 = 12.
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