In this lesson, I’ll cover some examples related to circular permutations.

** Example 1 **In how many ways can 6 people be seated at a round table?

**Solution **As discussed, the number of ways will be (6 – 1)!, or 120.

To the next..

** Example 2 **Find the number of ways in which 5 people A,B,C,D,E can be seated at a round table, such that

(i) A and B must always sit together.

(ii) C and D must not sit together.

**Solution (i) **If we wish to seat A and B together in all arrangements, we can consider these two as one unit, along with 3 others. So effectively we’ve to arrange 4 people in a circle, the number of ways being (4 – 1)! or 6. Let me show you the arrangements:

But in each of these arrangements, A and B can themselves interchange places in 2 ways. Here’s what I’m talking about:

Therefore, the total number of ways will be 6 x 2 = 12.

**(ii) **The number of ways in this case would be obtained by removing all those cases (from the total possible) in which C & D are together. The total number of ways will be (5 – 1)! or 24. Similar to (i) above, the number of cases in which C & D are seated together, will be 12. Therefore the required number of ways will be 24 – 12 = 12.

Another example related to seating…

** Example 3 **In how many ways can 3 men and 3 ladies be seated at around table such that no two men are seated together?

**Solution **Since we don’t want the men to be seated together, the only way to do this is to make the men and women sit alternately. We’ll first seat the 3 women, on alternate seats, which can be done in (3 – 1)! or 2 ways, as shown below. (We’re ignoring the other 3 seats for now)

Note that the following 6 arrangements are equivalent:

That is, if each of the women is shifted by a seat in any direction, the seating arrangement remains exactly the same. That is why we have only 2 arrangements, as shown in the previous figure.

Now that we’ve done this, the 3 men can be seated in the remaining seats in 3! or 6 ways. Note that we haven’t used the formula for circular arrangements now. This is so because, after the women are seated, shifting the each of the men by 2 seats, will give a different arrangement. After fixing the position of the women (same as ‘numbering’ the seats), the arrangement on the remaining seats is equivalent to a linear arrangement.

Therefore the total number of ways in this case will be 2! X 3! = 12.

I hope that you now have some idea about circular arrangements.

The next lesson will introduce you to combinations or selections.