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In my textbook it was given that the number of circular permutations of n different things taken r at a time (regarding anticlockwise and clockwise arrangements as different) is (nPr)/r. Any help regarding its proof will be appreciated.

Abcd
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    Sorry, what does "taken r at a time" mean here? Does it mean that we're permuting a subset of r items out of n? – user326210 Dec 05 '17 at 03:12
  • Yes like there are 15 people we have to seat at a round table having only 8 seats. – Abcd Dec 05 '17 at 03:14

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Suppose we're dealing with persons being assigned to chairs around a circular table. Suppose there are $n$ people and the chairs are numbered $1\ldots r$.

The first task is to pick an (ordered) list of $r$ people out of $n$ people to sit in the chairs— someone to sit in chair #1, #2, and so on. There are $nPr$ different seating assignments when considered this way.

But because the chairs are arranged in a circle, some of the different seating assignments are actually equivalent except for a rotation: a seating assignment like ABCDE should be considered the same as BCDEA and CDEAB, and DEABC, and EABCD.

In fact, for each circular permutation there are $r$ equivalent permutations where a different person sits in "Seat #1" while keeping the same seating order around the table. We divide by $r$ to avoid overcounting these variant permutations.

user326210
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