Where can I get 30 other people?

Discrete Mathematics Level 5

A game is played with \(31\) people.

First, each person is assigned a different integer between \(0\) and \(30\) inclusive. Also, an integer \(k\) is selected such that \(0 \leq k \leq 2015\).

Then, a computer program chooses a random subset of \(\{1, 2, 3, \dots, 2015 \}\) with \(k\) elements. (For \(k = 0\), the subset is the empty set.) Each subset is equally likely to be chosen. The sum \(S\) of the numbers in this subset is calculated.

The winner of the game is the person whose assigned number is congruent to \(S\) modulo \(31\).

It is given that there is one person out of all \(31\) people that has a greater probability of winning than the other \(30\) people. Find the sum of all possible values of \(k\) that satisfy this.


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