Q.4 A survey of 450 students about their subjects of interest resulted in the
following outcome.
150 students are interested in Mathematics.
200 students are interested in Physics.
175 students are interested in Chemistry.
50 students are interested in Mathematics and Physics.
60 students are interested in Physics and Chemistry.
40 students are interested in Mathematics and Chemistry.
30 students are interested in Mathematics, Physics and Chemistry.
Remaining students are interested in Humanities.
Based on the above information, the number of students interested in
Humanities is
(A) 10
(B) 30
(C) 40
(D) 45
The survey involves 450 students with specific interests in Mathematics (150), Physics (200), and Chemistry (175), including overlaps. Applying the inclusion-exclusion principle yields 405 students interested in at least one science subject, leaving 45 for Humanities.
Inclusion-Exclusion Formula
The principle for three sets |M∪P∪C| = |M| + |P| + |C| − |M∩P| − |P∩C| − |M∩C| + |M∩P∩C| avoids double-counting overlaps.
Substitute values: 150 + 200 + 175 − 50 − 60 − 40 + 30 = 405. Total science-interested students equal 405, so Humanities students = 450 − 405 = 45.
Venn Diagram Breakdown
Visualize regions:
- Only M:
150 − 50 − 40 + 30 = 90 - Only P:
200 − 50 − 60 + 30 = 120 - Only C:
175 − 60 − 40 + 30 = 105 - M∩P only:
50 − 30 = 20 - P∩C only:
60 − 30 = 30 - M∩C only:
40 − 30 = 10 - All three: 30
Sum confirms 405.
Option Analysis
- (A) 10: Too low; ignores correct subtraction yielding 45.
- (B) 30: Matches triple overlap but overlooks full union calculation.
- (C) 40: Equals M∩C; misapplies pairwise without inclusion-exclusion.
- (D) 45: Correct, as
450 − 405 = 45.
CSIR NET Application
This tests set theory for Life Sciences quantitative aptitude. Practice similar problems to master overlaps in genetics surveys or population studies.
