How many ways can 9 keys be arranged on a circular key ring?

To find the number of ways to arrange 9 keys on a circular key ring, we can use the concept of circular permutations.

The formula for the number of circular permutations of n objects is (n - 1)!. This is because when the objects are arranged in a circle, we can rotate the circle by any number of positions and obtain the same arrangement. So, we fix one object and arrange the remaining (n - 1) objects in a line.

In this case, we have 9 keys, so the number of circular permutations is (9 - 1)! = 8!.

To calculate 8!:
1. Start with the number 8.
2. Multiply it by the next lower number (7): 8 * 7 = 56.
3. Multiply the result by the next lower number (6): 56 * 6 = 336.
4. Continue this process until you reach 1: 336 * 5 * 4 * 3 * 2 * 1 = 40,320.

Therefore, there are 40,320 possible ways to arrange 9 keys on a circular key ring.

place 1 key as a marker, the rest of the keys can then be arranged in 8! ways,

so 40320 ways

question remaining, ...
does a 180 degree rotation of the key ring constitute a new ring?