Q. 12

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

Answer :

Given:

- Total number of people = 60

- Number of people who read newspaper H = 25

- Number of people who read newspaper T = 26

- Number of people who read newspaper I = 26

- Number of people who read newspaper H and I both = 9

- Number of people who read newspaper H and T both = 11

- Number of people who read newspaper T and I both = 8

- Number of people who read all three newspapers = 3

To Find:

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

Let us consider,

Number of people who read newspaper H = n(H)= 25

Number of people who read newspaper T = n(T) = 26

Number of people who read newspaper I = n(I) =26

Number of people who read newspaper H and I both =

n(H ∩ I ) = 9

Number of people who read newspaper H and T both =

n(H ∩ T ) = 11

Number of people who read newspaper T and I both =

n(T ∩ I ) = 8

Number of people who read all three newspapers =

n(H ∩ T ∩ I) = 3

Number of people who read at least one of the three newspapers= n(H Ս T Ս I)

Venn diagram:

We know that,

n(H Ս T Ս I) = n(H) + n(T) + n(I) – n(H ∩ I) – n(H ∩ T) –

n(T ∩ I) + n(H ∩ T ∩ I)

= 25 + 26 + 26 – 9 – 11 – 8 + 3

= 52

Therefore,

Number of people who read at least one of the three newspapers = 52

(ii) The number of people who read exactly one newspaper

Number of people who read exactly one newspaper =

n(H Ս T Ս I) – p – q – r – s

Where,

p = Number of people who read newspaper H and T but not I

q = Number of people who read newspaper H and I but not T

r = Number of people who read newspaper T and I but not H

s = Number of people who read all three newspapers = 3

p + s = n(H ∩ T) …(1)

q + s = n(H ∩ I) …(2)

r + s = n(T ∩ I) …(3)

Adding (1), (2) and (3)

p + s + q + s + r + s = n(H ∩ T) + n(H ∩ I) + n(T ∩ I)

p + q + r + 3s = 9 + 11 + 8

p + q + r + s + 2s = 28

p + q + r + s = 28 - 2×3

p + q + r + s = 22

Now,

n(H Ս T Ս I) – p – q – r – s = n(H Ս T Ս I) – (p + q + r + s)

= 52 -22

= 30

Hence, 30 people read exactly one newspaper.

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