Question

In how many ways can 6 persons be arranged in 

(i) a line, 

(ii) a circle?

Answer 1

(i) Let choose 1 person from 6 by ⁶C₁ = 6 and arranged it in line

Now choose another person from remaining 5 by ⁵C₁=5 and arranged it in line 

Similarly, choose another person from remaining 4 by ⁴C₁=4 and arranged it in line 

Similarly, choose another person from remaining 3 by ³C₁=3 and arranged it in line 

Similarly, choose another person from remaining 2 by ²C₁= 2 and arranged it in line 

And choose another person from remaining 1 by ¹C₁=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^st and 2^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)!