This problem is from the Test of Mathematics, TOMATO Subjective Problem no. 172 based on the Round Robin tournament.
Problem : Suppose there are teams playing a round robin tournament; that is, each team plays against all the other teams and no game ends in a draw.Suppose the team loses games and wins games. Show that
Solution : Each team plays exactly one match against each other team.
Consider the expression
Since each team plays exactly k-1 matches and no match ends in a draw, hence number of wins plus numbers of loses of a particular team is k-1 (that is the number of matches it has played). In other words for all i (from 1 to k).
But (as total number of loses = total number of matches = total number of wins; as each match results in a win or lose of some one)
Therefore implying =