2021 WSMO Speed Round Problems/Problem 8

Problem

Let $n$ be the number of ways to seat $12$ distinguishable people around a regular hexagon such that rotations do not matter (but reflections do), and two people are seated on each side (the order in which they are seated matters). Find the number of divisors of $n$ .

Solution

Note that there are $12!$ ways to seat the people. Since rotations are considered the same, we have to divide by 6. In addition, since the order in which the two people are seated on each side does not matter, we have to divide by $2^6=64.$ Thus, given the conditions, there are $\frac{12!}{64\cdot6}=2^3\cdot3^4\cdot5^2\cdot7^1\cdot11^1.$ In conclusion, the answer is $(3+1)(4+1)(2+1)(1+1)(1+1)=\boxed{240}.$

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2021 WSMO Speed Round Problems/Problem 8

Problem

Solution