Number of circular permutations of $n$ different things taken $r$ at a time. My book writes like this: > **Number of permutations of $n$ different things taken $r$ at a time** : **Case I : If clockwise & anti-clockwise orders are taken as different, then the number of circular permutations** = $$ \frac{{_n}P_r}{r}$$ **Case II : When clockwise & anticlockwise orders are not taken as different, then number of permutations** = $$\frac{{_n}P_r}{2r}$$ So, can anyone help me explaining how the book came to the conclusion? What is the proof? Please help.

nPr is the count of the number of arrangements of n unique beads taken r together. Once you arrange them, tie them into a necklace of r beads.

Now think of cutting the necklace. You can _choose_ to cut at r points. Each cut will result in a permutation of the beads. All these r permutations would result in the same necklace.

Out of the nPr permutations, there are r permutations for _each_ necklace. Hence, the number of _unique_ necklaces is nPr/r

Once we prove case I, it is easy to prove case II. Each unique necklace of case I, will also have its mirror reflection, which was also counted. Case II considers these 2 as same: hence the total count is halved.