Explanation: In this problem, we are dealing with the concept of overlapping sets, which is a fundamental topic in set theory and counting principles. The principle of inclusion-exclusion is a key concept used to solve problems involving overlapping sets. It helps us find the total number of unique elements in the union of two or more sets when there is an overlap between them.
Let's break down the problem step by step:
1. **Identify the Sets and Overlap:**
- Set A (Cricket Team): 8 students
- Set B (Badminton Team): 12 students
- Overlap (Students in both teams): 3 students
2. **Apply the Principle of Inclusion-Exclusion:**
The principle of inclusion-exclusion for two sets is given by the formula:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
where:
- \( |A \cup B| \) is the total number of unique students in both teams.
- \( |A| \) is the number of students in the cricket team.
- \( |B| \) is the number of students in the badminton team.
- \( |A \cap B| \) is the number of students in both teams.
3. **Substitute the Values:**
\[
|A \cup B| = 8 + 12 - 3 = 17
\]
4. **Interpret the Result:**
The total number of unique students in both teams is 17. This means that when we count the students in both teams, we must subtract the overlap to avoid double-counting the students who are in both teams.
Understanding the principle of inclusion-exclusion is crucial for solving problems involving overlapping sets. It ensures that we count each unique element only once, even when elements belong to multiple sets. This concept is widely applicable in various fields, including mathematics, statistics, and competitive examinations in India such as UPSC, SSC, Bank PO, CLAT, and MAT.
In summary, the correct answer is 17 because we add the number of students in each team and then subtract the number of students who are in both teams to avoid double-counting. This approach ensures that we accurately count the total number of unique students in both teams.