Suppose that n greaterthanorequalto 4 and that of n people a

Suppose that n greaterthanorequalto 4 and that of n people, any two of them together know all the remaining n - 2. Show that the people can be seated around a circular table so that everyone is seated between two friends.

Solution

In general, the number of ways of arranging n objects around a round table is (n-1)!

An easier way of thinking is that we \"fix\" the position of a particular person at the table. Then the remaining n -1 persons can be seated in (n-1)! ways. Done!

Thus the number of ways of arranging n persons along a round table so that no person has the same two neighbours is(n-1)!/2

Similarly in forming a necklace or a garland there is no distinction between a clockwise and anti clockwise direction because we can simply turn it over so that clockwise becomes anti clockwise and vice versa. Hence the number of necklaces formed with n beads of different colours = (n-1)!/2