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2 problems match “cycle decomposition”. Clear

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1. Permutations with order dividing twelve ●●○○○ 20 ptsLet $A=\{1,2,\dots,6\}$ and let $f: A \to A$ be a one-to-one mapping. Define $f^{(2)}(x)=f(f(x))$ and $f^{(n+1)}(x)=f(f^{(n)}(x))$ for $n \ge 1$. Find the number of mappings $f$ such that $f^{(12)}(x)=x$ for every $x \in A$.
2. Athletes taking someone else's ball ●●○○○ 20 ptsFive athletes came to training, each with their own ball. When leaving, each athlete took a ball that was not their own, and each ball was taken by exactly one athlete. In how many ways is this possible?