# In a survey of 60 people,

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In a survey of 60 people, it was found that 25 people read newspaper H,

11 read both H and T, 8 read both T and I,

3 read all three newspapers. Find:

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

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

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Let A be the set of people who read newspaper H.

Let  B be the set of people who read newspaper T.

Let C be the set of people who read newspaper I.

Accordingly,

n(A) = 25, n(B) = 26, n(C) = 26 ,

n(A ∩ C) = 9, n(A ∩ B) = 11,

n(B ∩ C) = 8 ,

n(A ∩ B ∩ C)

= 3 Let U be the set of people who took part in the survey.

(i) Accordingly,

n(A B C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)

= 25 + 26 + 26 – 11 – 8 – 9 + 3 = 52

Hence,

52 people read at least one of the newspapers.

(ii) Let a be the number of people who read newspapers H and T only. Let b denote the number of people who read newspapers I and H only.

Let c denote the number of people who read newspapers T and I only.

Let d denote the number of people who read all three newspapers.

Accordingly,

d = n(A ∩ B ∩ C) = 3

Now,

n(A ∩ B) = a + d

n(B ∩ C) = c + d

n(C ∩ A) = b + d

∴ a + d + c + d + b + d

= 11 + 8 + 9 = 28

⇒ a + b + c + d

= 28 – 2d = 28 – 6 = 22

Hence,

(52 – 22) = 30 people read exactly one newspaper.