How many ways can 8 people be seated around a circular table if two of them insist on sitting beside each?

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 in the lesson, the number of ways will be (6 – 1)!, or 120.

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

(i) A and B always sit together.
(ii) C and D never 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’s take a look at these 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 or 12.

(ii) The number of ways in this case would be obtained by removing all those cases (from the total possible) in which C and 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 and D are seated together, will be 12. Therefore the required number of ways will be 24 – 12 or 12.

Example 3 In how many ways can 3 men and 3 women be seated at around table such that no two men sit 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 woman shifts 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 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! or 12.

I hope that you now have some idea about circular arrangements. The next lesson will introduce you to combinations or selections.

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