Question

Find a formula for the number of circular\(r\)-permutations of\(n\)people.

Short answer

The required formula is \(\frac{{n!}}{{r(n - r)!}}\).

Step-by-step solution

Definition of Division rule and permutation

Division rule is if a finite set\(A\) is the union of pairwise disjoint subsets with\(d\)elements each, then\(n = \frac{{|A|}}{d}\)

Definition permutation (order is important): \(P(n,r) = \frac{{n!}}{{(n - r)!}}\)

Use permutation and division rule

The order of the individuals is important since a different order leads to different seating.

Thus we need to use the definition of permutation.

We need to select\(n\)of the\(r\)people:\(r - 1\)

Note that each seating arrangement has\(r\)permutations associated with it because moving every person\(1\)position (or\(2\)or\(3\)or ... or\(r - 1\)) to the right is the same as the original seating arrange, while these\(r - 1\)seating arrangements are separate permutations.

Use the quotient rule:

\(\begin{array}{c}|A| = \frac{{n!/(n - r)!}}{r}\\ = \frac{{n!}}{{r(n - r)!}}\end{array}\).