In how many ways can 6 persons be arranged in
(i) a line, (ii) a circle?
(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 1st and 2nd 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)!