Solution
= Solution
In the <complete graph> every pair of vertices is adjacent, so all $n$ colours assigned to vertices must be distinct. Choose colours successively, leaving one fewer choice at each step. Thus
$$
\boxed{P_{K_n}(k)=k(k-1)\cdots(k-n+1).}
$$
For nonnegative integer $k<n$ the product is zero, as required.