Question

What is circular permutation? give some examples also

Answer 1

See below:

Explanation:

A permutation is a method to calculate the number of ways we can take some number of objects and arranging them in some order. For instance, we can say that we have 5 books and we want to put them on a shelf. The number of ways we can arrange those 5 books is `5! = 120` ways.

We can also take a smaller grouping and arrange those. For instance, from the 5 books, we want to take a group of 3 of them and arrange them on a shelf. How many ways can we do that? That calculation is:

`5xx4xx3=60` different ways. The permutation formula allows us to find that number directly:

`P_(n,k)=(n!)/((n-k)!); n="population", k="picks"`

Note that we can express the two situations above as:

`P_(5,5)=(5!)/(0!)=5! = 120`

and

`P_(5,3)=(5!)/(2!)=(5xx4xx3xx2!)/(2!)=5xx4xx3=60`

So what is circular permutation?

Let's say that instead of books, we have 5 people and they are going to sit at a table in a restaurant. What's different here is that instead of being in a line, the people are in a circle, and so having person A sitting in seat 1 with person B in seat 2 on the right and person E on the other side is the same as that same arrangement with person A in seat 2 (with B in seat 3 and E in seat 1). And seat 3, and seat 4, and seat 5.

And so we end up with duplicates of the arrangements. On a circular table, ABCDE is the same as EABCD is the same as DEABC, and so on.

And so to eliminate the duplicates, we divide by the number of places. In this case, we get:

`(5!)/5=120/5=24`

Notice that with `5! = 5xx4xx3xx2xx1`, we can also express this as:

`(5!)/5=4! = 24`

Let's go one step further. We have 5 people who walk into a restaurant but only 3 are going to eat - two people will sit at the bar. In how many different ways can we arrange the 3 people at the table?

We know that in our permutation, we have `(5!)/(2!)=3! = 60`, so 60 ways to arrange the 5 people in groups of 3 at the table. We then need to work with the table arrangement and recognize that ABC is the same as CAB, so we divide by 3 - the number of seats at the table, to get 20.

We can list those out:

ABC, ACB
ABD, ADB
ABE, AEB
ACD, ADC
ACE, AEC
ADE, AED
BCD, BDC
BCE, BEC
BDE, BED
CDE, CED