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.
