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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.

Ishan Taneja
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  • 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. – S.C.B. May 08 '16 at 16:26
  • @MXYMXY I know what you mean.But still there should be a more clear explanation. – Ishan Taneja May 08 '16 at 16:27
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    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. – lulu May 08 '16 at 16:28
  • @lulu That seems a bit confusing.Will $A1$ not change his position. – Ishan Taneja May 08 '16 at 16:30
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    Equivalently, one of the English is the Queen, and one of the chairs is a throne. – André Nicolas May 08 '16 at 16:30
  • What does position mean? It's a circle so nobody has a well defined position, except in relation to the others. If we declare that $A_1$ is in position $1$ then we can define the other positions in, say, clockwise order. – lulu May 08 '16 at 16:31
  • Still not understanding,sorry. – Ishan Taneja May 08 '16 at 16:32
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    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. – lulu May 08 '16 at 16:34
  • @lulu Wait, so If seat the A's and after that I will be left with not a circular arrangement but sort of linear, is that somehow it? And also if identical 'objects' are placed ,say eggs and apples,will the circularity apply there for both? – Ishan Taneja May 08 '16 at 16:34
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    Well, almost. Wherever $A_1$ is sitting, that defines the start of the linear order. Thereafter it has to alternate Brit/American. – lulu May 08 '16 at 16:35
  • @lulu I edited previous comment,could you explain that bit also? – Ishan Taneja May 08 '16 at 16:37
  • If the objects in each group are identical (but still the same number of each) then there's only one pattern (alternating). Not sure what you nave in mind there. If the numbers aren't identical you have to look for rotational symmetries associated with each particular pattern. – lulu May 08 '16 at 16:41
  • @lulu I thought considering that example would clear up the current question for me, anyways I got it .thanks. – Ishan Taneja May 08 '16 at 16:44

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