counting question

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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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In how man ways can 8 cars be arranged such that there are exactly 4 cars between 2 specified cars.
|dw:1440017302219:dw|
  • phi
are the "2 specified cars" fixed in those "parking spots"? are we parking 8 cars or 6 cars?

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they are just fixed such that 4 cars are between them
|dw:1440017527636:dw| this diadram can also be possible
  • phi
for each "2 car" situation, there are 6! ways to park the other cars. if we call the 2 cars A and B, there are 3 ways to park A and B with 4 spaces between them and if we swap them, another 3 so 6*6! ways
how did u get red six ->\(\color{red}{6}*6!\)
  • phi
A _ _ _ _B_ _ _A_ _ _ _ B _ _ _ A _ _ _ _ B and then switch A and B for the other 3
oh i see thanks ,
can i count this one too B _ _A_ _ _ _ from right it has 4 distance
  • phi
you want exactly 4 cars between 2 specified cars.
oh so i cant count that
  • phi
no , that is exactly 2 cars between them, so it does not count
|dw:1440018099258:dw|
i got confused with that , earlier where number where used instead of cars in some previuos question
  • phi
it can get confusing. people hate this stuff... it makes their heads hurt.

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