Circular Permutations: Examples


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

Q1. In how many ways can 6 people be seated at a round table?

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

To the next..

Q2. 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.

Soln. (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:

pc7-circular-permutations-examples-1

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

pc7-circular-permutations-examples-2

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…

Q3. In how many ways can 3 men and 3 ladies be seated at around table such that no two men are seated together?

Soln. 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)

pc7-circular-permutations-examples-31

Note that the following 6 arrangements are equivalent:

pc7-circular-permutations-examples-4

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.

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