In how many ways can 6 persons be arranged in
Question:

In how many ways can 6 persons be arranged in

(i) a line,

(ii) a circle?

Solution:

(i) Let choose 1 person from 6 by ${ }^{6} \mathrm{C}_{1}=6$ and arranged it in line

Now choose another person from remaining 5 by ${ }^{5} C_{1}=5$ and arranged it in line

Similarly, choose another person from remaining 4 by ${ }^{4} C_{1}=4$ and arranged it in line

Similarly, choose another person from remaining 3 by ${ }^{3} C_{1}=3$ and arranged it in line

Similarly, choose another person from remaining 2 by ${ }^{2} C_{1}=2$ and arranged it in line

And choose another person from remaining 1 by ${ }^{1} C_{1}=1$ and arranged it in line

So total number of ways is $6 !=720$

(ii) It is the same as above, by converting line arrangement into the circle but you need to remove some arrangement

Let suppose 6 persons as $A, B, C, D, E, F$ you need to arrange this 6 persons into a circle.

First, we arranged 6 persons in line(number of ways $=6 !$ )

NOTE: A, B, C, D , E, F and B, C, D, E, F, A consider as a different line, but when we arranged this 2 combination in circle then it becomes same,

i.e. Let takes us an example we need to arrange $A, N, O, D, E$.

We arrange it as shown. When we rotate first one, then $1^{\text {st }}$ and $2^{\text {nd }}$ became identical and so on that’s why all 5 are identical, and we count it as 1

Now come back to our questions

So total number of arrangement is $(6-1) !=5 !=120$

NOTE: When you want to arrange n persons in circle then a total number of ways is  $n ! / n$,

i.e. Total number of ways $=(n-1) !$