3. There are 4 teams in a football tournament. Each team plays every other team exactly once. In each match, the winner gets 3 points and the loser gets 0 points. In the case of a draw, both teams get 1 point. After all matches have been played, which of the following total number of points is it impossible for any team to have obtained?
A 4
B 5
C 6
D 7
E 8
Solution
E

Each team plays exactly three matches. Hence the maximum number of points any team can obtain is $3 \times 3=9$. A draw only gets 1 point. Hence the next highest total number of points possible, from two wins and a draw, is $2 \times 3+1=7$. Therefore it is impossible to obtain 8 points.
(Note: totals of 4,5 and 6 points can be obtained by one win, one draw and one loss, one win and two draws and two wins and a loss respectively.)