In how many different ways (relative to each other) can 8

Question

In how many different ways (relative to each other) can 8 friends sit around a square table with 2 seats on each side of the table?

A.  2 * 8!

B.  2 * 7!

C.  \frac{8!}{4!}

D.  (8-1)!

E.  \frac{8!}{2}

JimGMATAnswers · Accepted Answer

Normally the formula for items in a circle is (n-1)! - in this case this is the 7!

It is multiplied by 2 because of the rest of the information. If you have the same arrangement of people you can have everyone shift one position to the right (or left) and they will be sitting next to a different person. With 2 shifts they will be back to sitting next to the original person.

Hope this helps.

KarishmaB · Answer

I gave the solution to this problem in the post after the post on circular arrangements: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/10 ... ts-part-i/ The answer is 2*7! It's explained in detail on the link given above.

chetan2u · Answer

Hi,
We can take this square as a circle but with one difference....
first, if it were a circle , the ways would be 7!..
now only difference is in each arrangement of 7!, any particular person would be sitting with two different person on an edge of the table
that is he would be sitting with the person on his left on one side of the square and with the one on his right in the next arrangement without shifting relatively to each other..
so we multiply each arrangement with 2.. ans 7!*2

pacifist85 · Answer

I am not sure I understand this comopletely.

For me, all 8 seats for the first person are the same.  But , if he sits on seat 1, then there are 7! remaining ways for the other people to be seated. If he seats on seat 2 (the other corner of that same side of the table) then there are 7! remaining ways for the rest of the people to be seated. The difference is that person one will be next to another person depending on which seat he chooses to sit on.

So, in this sense, 2*7! makes sense.

But, I don't get why 2-4-6-8 seats are the same and 1-3-5-7 similarly the same, creating the *2 in the response.

pacifist85 · Answer

OK... I just realised that the first person choosing seat 1 or 2 , and generally choosing 1-3-5-7 or 2-4-6-8 is actually exactly the same thing...

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In how many different ways (relative to each other) can 8

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