Question

After a hard-fought football game, it was reported that, of the 11 starting players, 8 hurt a hip, 6 hurt an arm, 5 hurt a knee, 3 hurt both a hip and an arm, 2 hurt both a hip and a knee, 1 hurt both an arm and a knee, and no one hurt all three. Comment on the accuracy of the report.

Short answer

The report seems incorrect, as the total number of hurt players (13) surpasses the total number of players (11) according to the Principle of Inclusion and Exclusion.

Step-by-step solution

Identify the Sets

Identify the three groups: 'Hip' with 8 players, 'Arm' with 6 players, and 'Knee' with 5 players. These correspond to sets A, B, and C respectively.

Identify the Intersections

Based on the problem, identify the intersections of these sets. We have: 'Hip and Arm' with 3 players, 'Hip and Knee' with 2 players, and 'Arm and Knee' with 1 player. These correspond to |A ∩ B|, |B ∩ C|, and |C ∩ A|, respectively. We know that nobody was in all three groups, so |A ∩ B ∩ C| = 0.

Apply Principle of Inclusion and Exclusion (PIE)

Apply the Principle of Inclusion and Exclusion to calculate the total number of players hurt: |A U B U C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |C ∩ A| + |A ∩ B ∩ C|. Substituting the given numbers: |A U B U C| = 8 + 6 + 5 - 3 - 2 - 1 + 0 = 13.

Compare with Total Players

The total number of players was Given as 11. However, according to the Principle of Inclusion and Exclusion, we calculated the total number of hurt players as 13, which surpasses the total number of players. Thus, the report seems inaccurate.