E. Two Round Dances
combinatorics; math
One day, \(n\) people (\(n\) is an even number) met on a plaza and made two round dances, each round dance consists of exactly \(\frac{n}{2}\) people. Your task is to find the number of ways \(n\) people can make two round dances if each round dance consists of exactly \(\frac{n}{2}\) people. Each person should belong to exactly one of these two round dances.Round dance is a dance circle consisting of \(1\) or more people. Two round dances are indistinguishable (equal) if one can be transformed to another by choosing the first participant. For example, round dances \([1, 3, 4, 2]\), \([4, 2, 1, 3]\) and \([2, 1, 3, 4]\) are indistinguishable.For example, if \(n=2\) then the number of ways is \(1\): one round dance consists of the first person and the second one of the second person.For example, if \(n=4\) then the number of ways is \(3\). Possible options: one round dance — \([1,2]\), another — \([3,4]\); one round dance — \([2,4]\), another — \([3,1]\); one round dance — \([4,1]\), another — \([3,2]\). Your task is to find the number of ways \(n\) people can make two round dances if each round dance consists of exactly \(\frac{n}{2}\) people.
The input contains one integer \(n\) (\(2 \le n \le 20\)), \(n\) is an even number.
Print one integer — the number of ways to make two round dances. It is guaranteed that the answer fits in the \(64\)-bit integer data type.