<p>You would use 6!, but you would take into account that if everyone got up and moved one seat to the right, you would still have the same arrangement.</p>
<p>So there are 6!, or 720 ways of arranging the people.</p>
<p>But again, if everyone got up and moved one to the right, you’d still have the same arrangement. And there are 6 possibilities. So, if the people were letters, then ABCDEF is the same as BCDEFA, CDEFAB, DEFABC, EFABCD, and FABCDE. There are 6 possibilities for each, so you’d divide 720/6 = 120, in order to take out all the repeats.</p>
<p>-You can look at the Calling All Math Geniuses Thread too.</p>