in a class of 35 students, 15 study Economics; 22 study Business studies and 14 study Accountancy. If 11 students study both Economics & Business studies: 8 study both Business studies & Accountancy and 5 study both Economics & Accountancy, if 5 students study none of these subjects, find the number of students who study (1) All the three subjects (ii) Exactly two subjects (iii) Only one subject.
Answers
Answer:
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Concept:
In mathematics, a set is a logically arranged group of items that can be represented in either set-builder or roster form. Sets are typically denoted by curly brackets; for instance, A = 1, 2, 3, 4 is a set. Check the set symbols here as well.
Set theory was created to describe the object collection. This is where you first learned about the classification of sets. The various kinds of sets, symbols, and operations are described by the set theory.
Given:
In a class of 35 students, 15 study Economics; 22 study Business studies and 14 study Accountancy. If 11 students study both Economics & Business studies: 8 study both Business studies & Accountancy and 5 study both Economics & Accountancy, if 5 students study none of these subjects
Find:
Find the number of students who study
(1) All the three subjects
(ii) Exactly two subjects
(iii) Only one subject.
Solution:
There are 35 students the class
So, n(U)=35
n(A)=15, n(B)=22 and n(C)=14
n'(A∪B∪C)=5
n(A∪B∪C)=n(U)-n'(A∪B∪C)
=35-5
=30
1) As per question,
n(A∪B∪C)=n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(A∩C)+n(A∩B∩C)
30 =15+22+14-8-8-5+n(A∩B∩C)
n(A∩B∩C)=35-30
=5
Therefore, the no. of students studing all the subjects is 5
2) As per question,
Using the formula,
Exactly two subjects=n(A∩B)+n(B∩C)+n(A∩C)-3n(A∩B∩C)
=8+8+5-3x5
=6
Therefore, the no. of students studing only two subjects is 5
3) As per question,
Using the formula,
Exactly one subjects=
n(A)+n(B)+n(C)-2n(A∩B)-2n(B∩C)-2n(A∩C)+3n(A∩B∩C)
=15+22+14-16-16-10+15
=24
Therefore, the no. of students studing only one subjects is 24
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