# When to use circular permutation vs when to use linear permutation?

I'm learning about permutations, both linear and circular.

For most questions, I am spoon-fed whether the problem should use circular or linear permutation to solve. IE I'm told it's a ferris wheel or a necklace. However in the real world, I'm having trouble differentiating exactly when to use each.

For example, arranging 4 people on a 6 seat bus. It's not inherently circular, but you can imagine the bus having two rows of single seats, and it can form a loop along which you can rotate people. So do I use circular in that case?

In most permutation cases I imagine "rotations" being possibilities. For example along a line:

1, 2, 3, 4, 5 ---> 5, 1, 2, 3, 4 ---> 4, 5, 1, 2, 3

Etc.

So I'm trying to get some clarity on how, in the real world, you know which formula to use.

Thank you very much for any advice you may provide! :)

The difference is that in linear permuatations you have places laid out in (roughly) a line and in circular permuatations you have them in a circle such that two (circular) permuataions are considered the identical if you can get to one by rotating the other around the circle. Therefore if you have $n$ places to put things in then there will be $n$ times as many circular permutations as linear ones.