Now if we were to arrange them on a bench then we would use following formula n!
i.e; 1st position can be taken by any 1 of the 5 knights, 2nd position can be taken by only 1 of the 4 knights because 1 knight has already taken the 1st position, simillarly 3rd position can be taken by any 1 of the 3 knights .......
Thus the formula is = no of ways the 1st position can be filled *no of ways the 2nd position can be filled *no of ways the 3rd position can be filled * no of ways the 4th position can be filled *no of ways the 5th position can be filled = 5*4*3*2*1 = 120
But in circular permutation for distinct objects
one has to elliminate all the repeations i.e; we must count all the possibilities only once but we can take both clockwise and anticlockwise as seperate arrangements
Let X,Y,Z,P and Q be the knights
Here , x y z p q is a possible arrangement (clockwise) and x q p z y is also a possible arrangement (anticlockwise)
But y z p q x or z p q x y or q p z y x etc.. are not distinct arrangements.
Simillarly for ,
Thus we have the formula (n-1)!,
which elliminates all repeatitions.
But for not distinct objects
you have only one possible combination i,e; either clockwise or anticlockwise e.g. pearls in a necklace.
Thus we have the formula (n-1)!/2
Hope this helps