Explanations:In this problem, we are dealing with the concept of overlapping sets, which is a fundamental principle in set theory. The problem involves two sets: the cricket team and the badminton team. The key point is that there are students who are members of both teams, and we need to ensure that these students are not counted twice when determining the total number of students involved in either team.
To solve this problem, we use the principle of inclusion-exclusion. This principle states that to find the total number of elements in two overlapping sets, we add the number of elements in each set and then subtract the number of elements that are in both sets. This subtraction is necessary to avoid double-counting the students who are on both teams.
Let's break down the problem step-by-step:
1. Identify the number of students in each team:
- Cricket team: 8 students
- Badminton team: 12 students
2. Identify the number of students who are on both teams:
- Students on both teams: 3 students
3. Apply the principle of inclusion-exclusion:
- Total number of students = (Number of students in cricket team) + (Number of students in badminton team) - (Number of students on both teams)
- Total number of students = 8 + 12 - 3 = 17
Thus, the total number of students on the two teams, accounting for the overlap, is 17. This is why option A is the correct answer.
It is important to understand that without the principle of inclusion-exclusion, we would incorrectly count the students who are on both teams twice. This would lead to an overcount, which is why options B, C, and D are incorrect.
The principle of inclusion-exclusion is a powerful tool in set theory and combinatorics, and it is often used in various real-world scenarios where overlapping sets need to be counted accurately. Understanding this principle is crucial for solving problems involving overlapping sets in mathematics and other fields.