Solving the Venn Diagram Problem: Counting Students Who Failed in Exactly One Subject

Let's consider a challenging problem in set theory and Venn diagram representation. We have 131 students in a class, and we are given some information on how many students failed in different subjects:

Problem Statement

32 students failed in Economics.
30 students failed in Politics.
46 students failed in History.
12 students failed in both Economics and Politics.
9 students failed in both Politics and History.
10 students failed in both Economics and History.
3 students failed in all three subjects.

The question is: How many students failed in exactly one subject?

Understanding Venn Diagrams and Set Theory

A Venn diagram is a powerful tool for visualizing the relationships between sets. Here, we use three overlapping circles to represent students who failed in Economics (E), Politics (P), and History (H).

Formulate the Venn Diagram

Given the Venn diagram representation, the areas where circles overlap represent students who failed in multiple subjects. Here's the breakdown of the overlying areas:

The area representing students who failed in E and P but not H. The area for students who failed in P and H but not E. The area for students who failed in E and H but not P. The central area where E, P, and H all intersect.

Apply the Principle of Inclusion-Exclusion

To solve for students who failed in exactly one subject, we use the principle of inclusion and exclusion. The formula for the number of students failing in at least one subject is:

Sum of students failing in each subject. Subtract the sum of students failing in exactly two subjects. Add back the students failing in all three subjects.

The formula for exactly one subject is given by:

Exactly one subject (Students failing in E or P or H) - 2 * (Students failing in E and P or P and H or E and H) 3 * (Students failing in all three)

Solving the Equation

Substitute the given values:

Exactly one subject 32 30 46 - 2 * (12 9 10) 3 * 3

Calculate each term step-by-step:

Exactly one subject 108 - 2 * 31 9

Exactly one subject 108 - 62 9

Exactly one subject 55 9

Exactly one subject 75

Therefore, the number of students who failed in exactly one subject is 75.

Conclusion

Using the principle of inclusion and exclusion effectively helps solve complex set theory problems, such as the one presented. Understanding Venn diagrams and how to apply these principles can provide a clearer insight into solving similar problems in the future.