Answer to Question #279368 in Discrete Mathematics for Jaishree

Question

Consider a tournament with n players where each player plays against every

other player. Suppose each player wins at least once. Show that at least 2 of the

players have the same number of wins.

Answer 1

The number of wins for a player is at least $1$ and at the most $(n-1).$

These $(n-1)$ numbers correspond to the pigeonholes to accomodate $n$ playesr (pigeons).

Thus, by the Pigeonhole Principle there must be at least two players occupying one pigeonhole (number of wins).

Hence, it is proved.