Question

A circular\(r\)-permutation of\(n\)people is a seating of\(r\)of these\(n\)people around a circular table, where seatings are considered to be the same if they can be obtained from each other by rotating the table.

Find the number of circular -3 permutations of\(5\)people.

Short answer

The number of circular-3 permutations of 5 people are\(20\).

Step-by-step solution

Definition of Division rule, permutation and combination

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\(3\)of the\(5\)people:\(P(5,3) = \frac{{5!}}{{(5 - 3)!}} = \frac{{5!}}{{2!}} = 5 \cdot 4 \cdot 3 = 60\)

Each seating arrangement has 3 permutations associated with it because moving every person 1 position to the right is the same as the original seating arrange and similarly moving every person 2 positions to the right is the same as the original seating arrange, while these two seating arrangements are separate permutations.

Use the quotient rule: \(|A| = \frac{{60}}{3} = 20\).