16 persons sit at a round table with 38 seats. How many possible arrangements are there if arrangements that are similar with rotation are counted only once?
2026-04-30 05:00:50.1777525250
Permutation of Circular Arrangements with Rotation
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In any given seating arrangement, person.1 has to be sitting somewhere. We'll call that position position.1 (because of rotational symmetry, we don't care what seat they actually take...) Then we call the seat right of him position.2, and to the right of that position.3, etc. So now you have 37 ordered seats with 15 people left to fill them. There are $15!$ possible orders of the people, and for each order, there are $\binom{37}{15}$ possible spacing arrangements.
Thus, you have $ 15! \cdot \binom{37}{15}$ possible combinations.