There are 1500 students in a college. Each student can be a...

There are 1500 students in a college. Each student can be a member of three student communities namely P, Q, and R. Now using the data mentioned and the diagram given below answer the questions that follow. - Total members in community $\mathrm{P}$ is 300 . - Total members in community $\mathrm{Q}$ is 420 . - Total members in community $\mathrm{R}$ is 490 . How many students are part of at least two communities? (A) 10 (B) 30 (C) 80 (D) 90

There are 1500 students in a college. Each student can be a member of three student communities namely P, Q, and R. Now using the data mentioned and the diagram given below answer the questions that follow.
- Total members in community $\mathrm{P}$ is 300 .
- Total members in community $\mathrm{Q}$ is 420 .
- Total members in community $\mathrm{R}$ is 490 .
How many students are part of at least two communities?
(A) 10
(B) 30
(C) 80
(D) 90

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Key Concepts

Inclusion-Exclusion Principle

This principle is a fundamental counting technique in set theory that helps to accurately count elements in the union of multiple overlapping sets. It involves adding the sizes of individual sets, subtracting the sizes of all pairwise intersections to correct for double-counting, and then adding back the size of the intersection of all sets if necessary, to ensure each element is counted exactly once.

Venn Diagrams

Venn diagrams provide a visual method to represent sets and their overlaps. They allow for clear visualization of the relationships between groups, including intersections and unions, and are especially useful in problems that require partitioning a universal set into segments based on membership in multiple subsets.

Set Theory and Overlapping Sets

Understanding set theory is crucial for dealing with collections that may share common elements. When various groups overlap, it is important to account for elements that belong to more than one group. This concept underpins strategies for avoiding overcounting and is essential in problems involving multiple group memberships.

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