[Math] eight people around square table

combinatoricspermutations

I would like you to help me with one example.
There are eight people(A,B,…,H). In how many ways can eight people, denoted A,B,…..,H be seated around the square table? The result is $10,080$ possible ways. I've also found three possible ways how to solve this example but I still can't get it.

  1. The first method is 8!/4. Why do we have to divide by $4$ ?

  2. The second method is 7! x 2. Why do we have to multiply by $2$ ?

  3. The last method is following: In how many ways can i split these eight people to four groups of two?
    The last method is clear to me, but I do not know how to calculate it. Can you help me ?

seating_arrangements_at_square_table

Best Answer

  1. A square table has four sides, and evidently we are considering rotations of a solution to be the "same" solution.

  2. We pick one special person to sit down on the north side of the table. Now rotations are no longer relevant, so there are $7!$ ways to seat everyone else. We multiply by $2$ because our special person has two places to sit on the north side.

  3. This is a different question than the previous ones, because the pairs of people now need to sit at the table, which they can do in different ways.

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