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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! :)

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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.

If in your real life situation you consider things the same if one can be rotated into the other then you should be counting with circular permuatations. Otherwise use linear permutations.

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