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Question

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 reas newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspaper,

Find :

(i) the numbers of people who read at least one of the newspapers.

(ii) the numbers of people who read exactly one newspaper.

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Solution

(i) Let

n(P) denote total number of people

n(H) denote number of people who read newspaper H

n(T) denote number of people who read newspaper T and

n(I) denote number of people who read newspaper I

Then, n(P) = 60, n(H) = 25, n(T) = 26, n(I) = 26

n(H∩I)=9,n(H∩T)=11,n(T∩1)=8,n(H∩T∩I)=3

We need to find the number of people who read atleast one of the newspaper, i.e., n(H or T or I), i.e., n(H∪T∪I), i.e., n(H∪T∪I) we know that if A, B, C are 3 sets , then,

n(A∪B∪C)=n(A)+n(B)+n(C)−

n(A∩B)−n(B∩C)−n(A∩C)+n(A∩B∩C)

n(H∪T∪I)=n(H)+n(T)+n(I)−

n(H∩T)−n(T∩I)+n(H∩T∩I)

= 25+26+26-9-11-8+3

= 25+52-28+3

= 25+52-25

=52

Hence, 52 people read at least one of the newspaper.

(ii) Now, we have to calculate the number of people who read exactly one newspaper.

We have,

n(H)+n(T)+n(I)−2n(H∩T)−

2n(T∩I)−2n(H∩I)+3n(H∩T∩I)

= 25+26+26-22-16-18+9= 30

Thus, 30 people can read exactly one newspaper.


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