Question 1138961
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You have the correct line of thinking and the correct final answer. 


You basically have seven letters
A,B,C,D,E,F,G
such that six of the letters are individual people while the seventh is composed of two people as one unit. For any permutation of the seven items, you can swap the two people sitting together, hence the "times 2" portion. Basically this is paraphrasing your awesome explanation you posted.


The line of thinking that you would go with 6! is not fully correct because it only considers arranging {A,B,C,D,E,F,G} such that one letter (say the letter A) stays in one spot and the other six permute or move around. You have to consider the letter A moving around as well to get the full picture. I can see why they went with 6! because 8-2 = 6 people are free to move around as opposed to the 2 individuals who must stay together. If it was a case where the chair and vice chair people must be in the first two slots, then the answer would be 2*6! = 2*720 = 1440; however this is not the case.


So once again, the <font color=red>answer of 10,080 is correct</font>. Nice work on getting the correct answer. Also I appreciate you posting your thoughts and your attempts at the answer, rather than just post the problem all by itself. 
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