How many different ways are there to seat 6 people at a round table?

First, let's think of how many different possibilities we have to seat 6 people at a 'normal' table, i.e. in a straight line. Let's call the people A, B, C, D, E, F. We now have 6 seats:

__   __   __   __   __

When we pick who gets to sit on the first seat, we have 6 options to choose from. For the second seat, there are still 5 people left to choose from, 4 for the third, 3 for the fourth, 2 for the fifth and only one person is left over to sit on the last seat. So we have:

6 x 5 x 4 x 3 x 2 x 1 = 6! = 720 possibilities.

Now if instead the table is round, there is 'no first seat'. If everybody rotates through one seat to the left, each person still has the same people to their left and right and we say the configuration is identical. There are 6 different seats for A to seat on that correspond to identical configurations (simply keep rotating the whole group through) - so compared to the 'normal' table, we have overcounted by a factor of 6. The number of possibilities is now:

6! / 6 = 720 / 6 = 120

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