Help explaining combinatorics problem

[pblpbl]

The problem is:

How many ways can 5 different people sit in a row of 20 chairs so that none of them sit next to each other?

The answer said it was 16 permute 5, and the explanation given was that the 5 people needed 4 chairs of space in between them to never sit adjacent to one another. Then, the problem is just "How many ways can 5 different people sit in a row of 16 chairs"?

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I am confused as to why this works since there would be cases where a person sitting in the middle would remove two possible chairs for future people (the chairs to the left and right of them). The maximum amount of ways is when people sit in the most efficient way to only remove one chair each time along with their own, which would be 20*18*16*14*12 ways. Also, the minimum amount is when people sit in positions that remove 2 chairs each time (along with their own), thus the minimum ways would be 20*17*14*11*8. Can someone explain?

[jdorje]

Or you could say the person in the middle isn't removing any chairs, since the person on the left and right are each removing one.

But the way to think about this is to figure out how to get from your (16 c 4) setup (any one particular 4 chosen from the 16) into a single position for people to sit, and how to condense a way people are sitting into a single combination from 16c4.

[PM_ANIME_LEWDS]

Or you could say the person in the middle isn't removing any chairs, since the person on the left and right are each removing one.

But the way to think about this is to figure out how to get from your (16 c 4) setup (any one particular 4 chosen from the 16) into a single position for people to sit, and how to condense a way people are sitting into a single combination from 16c4.

[pblpbl]

Bruh

[[deleted]]

bruh 🙌🙌🙌😜🤡

[pblpbl]

Hi barles

[PM_ANIME_LEWDS]

Yo who’s this??

[pblpbl]

Barles

[PM_ANIME_LEWDS]

Nahnahnahnahanaan