Out of a group of 60 students, 25 play cricket, 30 play football, and 24 play hockey. 10 students play both cricket and football, 9 play both cricket and hockey, 12 play both hockey and football, and 5 play all three games. Using a Venn diagram, find the number of students who play only one game.

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Points to Remember:

  • Understanding set theory and Venn diagrams is crucial.
  • The principle of inclusion-exclusion is vital for solving this problem.
  • Careful calculation and attention to detail are necessary to avoid errors.

Introduction:

This question involves the application of set theory to determine the number of students playing only one sport out of three offered: cricket, football, and hockey. The problem requires using a Venn diagram to visually represent the overlapping sets of students participating in different sports and then applying the principle of inclusion-exclusion to calculate the number of students involved in only one sport. Such problems are common in statistics and probability, helping us understand how to analyze overlapping data sets.

Body:

1. Visual Representation using a Venn Diagram:

We can represent the data using a Venn diagram with three overlapping circles, one for each sport: cricket (C), football (F), and hockey (H).

[Unfortunately, I can’t create visual diagrams here. Imagine a Venn diagram with three overlapping circles representing C, F, and H. The overlapping regions represent students playing combinations of sports.]

2. Filling the Venn Diagram:

We start by filling in the intersection of all three sets (C ∩ F ∩ H): 5 students.

Next, we fill in the intersections of two sets:

  • C ∩ F (but not H): 10 – 5 = 5 students
  • C ∩ H (but not F): 9 – 5 = 4 students
  • F ∩ H (but not C): 12 – 5 = 7 students

Now, we find the number of students playing only one sport:

  • Only Cricket (C): 25 – (5 + 5 + 4) = 11 students
  • Only Football (F): 30 – (5 + 5 + 7) = 13 students
  • Only Hockey (H): 24 – (4 + 5 + 7) = 8 students

3. Calculation of Students Playing Only One Game:

The number of students playing only one game is the sum of students playing only cricket, only football, and only hockey:

11 (only cricket) + 13 (only football) + 8 (only hockey) = 32 students

Conclusion:

In summary, by using a Venn diagram and applying the principle of inclusion-exclusion, we determined that 32 out of 60 students play only one of the three sports offered (cricket, football, or hockey). This approach highlights the importance of visual aids in solving problems involving overlapping sets. For future analysis of student participation in extracurricular activities, a similar approach could be used to understand preferences and allocate resources effectively. This method can be extended to analyze participation in more than three activities, providing valuable insights for school administration in optimizing sports programs and student engagement. The use of such analytical techniques promotes a data-driven approach to decision-making, leading to more holistic and effective resource allocation within the educational setting.

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