ILL 1986 Problem 39

Let $S$ be a $k$ -element set.

(a) Find the number of mappings $f : S \to S$ such that $$\text{(i) } f(x) \neq x \text{ for } x \in S, \text{(ii) } f(f(x)) = x \text{ for }x \in S.$$

(b) The same with the condition $\text{(i)}$ left out.