 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/r/r2/d/ii/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

P_{K_{n}} (k) = k (k - 1) \dots (k - n + 1) .

For nonnegative integer

k < n

the product is zero, as required.

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