We define and enumerate circular permutations.

Circular Permutations

In a circular permutation, all positions on the circle are considered equivalent. Thus, the position of an object is solely determined by its position relative to the other objects. By contrast, the objects in ordinary permutations have absolute positions- first, second, third etc. An ordinary permutation can be thought of as a linear permutation.

First, we select the objects to be placed in the circular permutation. This can be done ways.
Second, we arrange the objects in a circle and use the FPC. When the first object is placed in the circle, all of the positions are equivalent, so there is only 1 choice. Once the first object is placed, the remaining positions in the circle become distinct. Hence, there are choices for the position of the second object in the circle, choices for the third object, and so on. Thus, the number of ways to arrange the objects in a circle is . Finally, we multiply the number of ways to select the objects by the number of ways to arrange them in a circle to obtain To see that this is equal to , we note that

Applying the proposition with yields that the number of such permutations is
(problem 1) In how many ways can the 20 members of the high-school band march in a circle during the halftime show?
(problem 2) In how many ways can the President, Vice-President, Secretary of State, Secretary of the Treasury, Secretary of the Interior, Secretary of Labor and a secretary be seated at a round table if the President and Vice-President cannot be seated next to each other? .
(problem 3) In how many ways can 8 teenagers sit at a round table if the 4 boys sit next to each other?
(problem 4) Fifteen children run to get in line to get on the carousel when the local amusement park opens its doors at 10am. The ride has 12 seats arranged in a circle. The seats alternate horse, horse, tiger, tiger, etc. How many ways are there for the first group of 12 children to be seated on the ride?
(problem 5) Jean has 6 keys to put on a key ring, but does not want to put the house key next to the office key. In how many ways can Jean configure the key ring?