Chess Word Problem

CHESS WORD PROBLEM
Six students were in a chess tournament in which
each played each of the others once.
A player got 2 points for a win,
1 point for a draw, and
 0 points if they lost.
Here are the final points received after all the matches:
Adam - 4, Bert - 7, Conrad - 9, Dora - 5, Ernie - 1, Fred - ???
How many points did Fred get?
***STOP HERE UNLESS YOU WANT TO SEE THE SOLUTION***
THE SOLUTION IS :
******************SOLUTION*******************
There were 15 games played.
6 choose 2 results in 15 games.
Each type of game will result in 2 points awarded:
- 2 for winner AND 0 for loser       or
- 1 point for each player in tie games.
Thus the final scores must total 30 points.
Since the five known point values total 26,
Fred had 4 points.
In fact, since 4 people had an ODD number of points,
there must have been at least 4 games that ended in a tie!
FRED HAD    4 points