Five friends decided to play a game of badminton. Each of the five playsagainst everyother friends. The winner gets two points for each game he or she wins and the loser gets zero . Then which of the following cannot represent the scores of five friends ?

Options

A. 4, 4, 4, 4, 4
B. 6, 4, 4, 4, 2
C. 8, 8, 2, 2, 0
D. 6, 6, 4, 2, 2

Answer

To solve the problem, we need to analyze the scoring system in the badminton game played by five friends. Each player plays against every other player, and the winner of each game receives 2 points while the loser receives 0 points.

### Step-by-Step Solution:

1. **Determine the Total Number of Matches**:
Each player plays against every other player. For 5 players, the total number of matches can be calculated using the combination formula \( C(n, 2) \), where \( n \) is the number of players:
\[ \text{Total Matches} = C(5, 2) = \frac{5 \times 4}{2 \times 1} = 10 \]

2. **Calculate Total Points Distributed**:
In each match, a total of 2 points is awarded (2 points to the winner and 0 to the loser). Therefore, for 10 matches, the total points distributed will be:
\[ \text{Total Points} = 10 \times 2 = 20 \]

3. **Understand the Points Distribution**:
Each player can earn points based on the number of matches they win. The maximum points a player can earn is 8 points (if they win all 4 matches against the other players).

4. **Possible Scores for Players**:
Since there are 5 players, the total points scored by all players must equal 20. This means the sum of the scores of all players must equal 20.

5. **Check Possible Score Combinations**:
Let's denote the scores of the players as \( P_1, P_2, P_3, P_4, P_5 \). We need to find combinations of these scores such that:
\[ P_1 + P_2 + P_3 + P_4 + P_5 = 20 \]
Each score must be a multiple of 2 (since each win gives 2 points).

6. **Identify Invalid Score Combinations**:
We need to check which of the given score combinations cannot satisfy the above conditions. For example, if one of the scores is 8, then the remaining scores must sum to 12, and they must also be even numbers.

7. **Conclusion**:
After checking the possible combinations, we find that if any score exceeds 8 or if the total does not sum to 20, that combination cannot represent the scores of the five friends.