Let
be a prime number. For any permutation
of the set
, let
denote the number of multiples of
among the following
numbers:
Determine the average value of
taken over all permutations
of
.
%V0
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 $S$.
Školjka