You are arranging the integers from 1 to 12 in a circle like

You are arranging the integers from 1 to 12 in a circle (like on a clock). We\'ll consider two arrangements the same if they only differ by a rotation a) How many different ways are there to arrange these numbers with no restrictions? b) How many different ways are there to do this so that even and odd numbers alter- nate? (c) How many different ways are there to do this so that the \'1\' and the 12 are not next to each other?

Solution

There are (n1)! ways to arrange n distinct objects in a circle (where the clockwise and anti-clockwise arrangements are regarded as distinct.

a) Without any restrictions these 1 to 12 integers can be arranged in a circle is (12 - 1)! or 11!

B) We’ll first seat the 6 even numbers, on alternate seats, which can be done in (6 – 1)! or 5! ways (We’re ignoring the other 6 seats for now).These remaining these 6 seats can be seated by 6 odd numbers by 6! ways.

So , the number of ways of placing 6 even and 6 odd numbers in a circle is 5! x 6!.

C) First we set 1 and 12 as a single unit then these 11 numbers seated in a circle so that 1 and 12 sit together is (11-1)! or 10! ways. we can arrange 1 and 12 numbers in between them in 2! ways.

So, the number of ways of placing these 12 numbers such that 1 and 12 sit together is 2! x 10!