MEMO 2012 T4

Let $p>2$ be a prime number. For any permutation $\pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) )$ of the set $S = \{ 1, 2, \cdots , p \}$ , let $f( \pi )$ denote the number of multiples of $p$ among the following $p$ numbers: $$\pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p)$$ Determine the average value of $f( \pi)$ taken over all permutations $\pi$ of $S$ .