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 is not part of any community? (A) 390 (B) 420 (C) 410 (D) $4 \overline{90}$

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 is not part of any community?
(A) 390
(B) 420
(C) 410
(D) $4 \overline{90}$

Key Concepts

Complement of a Set

The complement of a set includes all the elements in the universal set that are not in the specified set. In the context of the question, after determining the total number of students part of any community (i.e., the union of the communities), the complement represents those students who do not belong to any community. This concept is essential for solving problems where the number of elements outside the defined groups is required.

Venn Diagram

A Venn diagram is a visual tool that represents sets as overlapping circles, making it easier to understand the relationships between them, including intersections and unions. It helps in organizing information about subsets and their intersections, which is particularly useful when applying the inclusion-exclusion principle in counting problems involving multiple overlapping communities.

Inclusion-Exclusion Principle

This principle is a fundamental counting technique used in set theory to calculate the total number of elements in the union of 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 triple (or higher) intersections if necessary. In problems involving multiple groups, such as membership in student communities, the inclusion-exclusion principle ensures every element is counted only once.

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