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The question is 4 Americans and 4 English are seated on a round table.No Two Americans sit together.Find the number of ways.

So,after this I did: $(4-1)!$ for seating the Americans around the table.And the book says afterwards that the Englishmen can be seated in $4!$ ways

I want to know that when 4 Americans are seated ,we take circular permutations but why not so for the englishmen. I mean even though we restrict the number of options after seating the Americans,shouldn't there be circular arrangement for the Englishmen.? Just a small conceptual doubt.

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  • $\begingroup$ I probably am not the greatest answerer out there, so I would add this as a comment, but realize now that Americans have been seated, the table can no longer be rotated and be the same. And now I read my comment and realizes that it sounds confusing. I would hope that I had access to some visual tools but unfortunately, I do not. $\endgroup$
    – S.C.B.
    May 8, 2016 at 16:26
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    $\begingroup$ Since we can rotate the table, we can always imagine that one particular individual, let's say $A_1$, is in position #$1$. Then there are three remaining slots for the Americans and four for the Brits. If you preferred, you could "start the order" from $E_1$ in which case there would be three slots for the remaining Brits, and four for the Americans. $\endgroup$
    – lulu
    May 8, 2016 at 16:28
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    $\begingroup$ Equivalently, one of the English is the Queen, and one of the chairs is a throne. $\endgroup$ May 8, 2016 at 16:30
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    $\begingroup$ Ok. The way I give my direction to the guests is "let Bob sit first. That'll be position $1$. Then position $2$ will be next to him clockwise and so on." In that way I can just assign the numbers $\{2,3,4,5,6,7,8\}$ to the other guests and that will suffice to describe an arrangement. $\endgroup$
    – lulu
    May 8, 2016 at 16:34
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    $\begingroup$ Well, almost. Wherever $A_1$ is sitting, that defines the start of the linear order. Thereafter it has to alternate Brit/American. $\endgroup$
    – lulu
    May 8, 2016 at 16:35

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