In a survey of 60 persons, it is found that 25 read newspaper H, 26 read T, 26 read I, 9 read H and I, 11 and H and T,8 read T and I and 3 read all three newspaper. Find :
(a) No. of persons who read atleast one newspaper.
(b) No. of persons who read exactly one newspaper.

Answer

To solve the problem, we will use the principle of inclusion-exclusion to find the number of persons who read at least one newspaper and the number of persons who read exactly one newspaper.

### Given Data:
- Total persons surveyed = 60
- Persons reading newspaper H (N(H)) = 25
- Persons reading newspaper T (N(T)) = 26
- Persons reading newspaper I (N(I)) = 26
- Persons reading both H and I (N(H ∩ I)) = 9
- Persons reading both H and T (N(H ∩ T)) = 11
- Persons reading both T and I (N(T ∩ I)) = 8
- Persons reading all three newspapers (N(H ∩ T ∩ I)) = 3

### Part (a): Number of persons who read at least one newspaper

We need to find N(H ∪ T ∪ I), which represents the number of persons who read at least one newspaper. The formula for this is:

\[ N(H \cup T \cup I) = N(H) + N(T) + N(I) - N(H \cap T) - N(H \cap I) - N(T \cap I) + N(H \cap T \cap I) \]

Substituting the values we have:

\[ N(H \cup T \cup I) = 25 + 26 + 26 - 11 - 9 - 8 + 3 \]

Calculating step by step:

1. Sum of persons reading each newspaper:
\[ 25 + 26 + 26 = 77 \]

2. Sum of persons reading two newspapers:
\[ 11 + 9 + 8 = 28 \]

3. Adding persons reading all three newspapers:
\[ 77 - 28 + 3 = 52 \]

Thus, the number of persons who read at least one newspaper is **52**.

### Part (b): Number of persons who read exactly one newspaper

To find the number of persons who read exactly one newspaper, we need to calculate the number of persons reading only H, only T, and only I.

Let:
- A = Persons reading all three newspapers = 3
- B = Persons reading H and T but not I
- C = Persons reading H and I but not T
- D = Persons reading T and I but not H
- E = Persons reading only H
- F = Persons reading only T
- G = Persons reading only I

From the given data, we can express:
- N(H) = E + B + C + A
- N(T) = F + B + D + A
- N(I) = G + C + D + A

Now substituting the known values:

1. For H:
\[ 25 = E + B + C + 3 \]
\[ E + B + C = 22 \quad \text{(1)} \]

2. For T:
\[ 26 = F + B + D + 3 \]
\[ F + B + D = 23 \quad \text{(2)} \]

3. For I:
\[ 26 = G + C + D + 3 \]
\[ G + C + D = 23 \quad \text{(3)} \]

Now we also know:
- B + C = 9 (from H and I)
- B + D = 11 (from H and T)
- C + D = 8 (from T and I)

From these equations, we can find the values of B, C, and D.

1. From B + C = 9 and B + D = 11:
\[ D - C = 2 \quad \text{(4)} \]

2. From B + C = 9 and C + D = 8:
\[ B - D = 1 \quad \text{(5)} \]

Now we can solve these equations step by step.

Using equations (4) and (5):
- From (4): \( D = C + 2 \)
- Substitute in (5): \( B - (C + 2) = 1 \)
- Thus, \( B - C = 3 \) or \( B = C + 3 \)

Substituting B into (1):
\[ E + (C + 3) + C = 22 \]
\[ E + 2C + 3 = 22 \]
\[ E + 2C = 19 \quad \text{(6)} \]

Now substituting B into (2):
\[ F + (C + 3) + D = 23 \]
Using \( D = C + 2 \):
\[ F + (C + 3) + (C + 2) = 23 \]
\[ F + 2C + 5 = 23 \]
\[ F + 2C = 18 \quad \text{(7)} \]

Now we have two equations (6) and (7):
1. \( E + 2C = 19 \)
2. \( F + 2C = 18 \)

From (6):
\[ E = 19 - 2C \]
From (7):
\[ F = 18 - 2C \]

Now we can find G using equation (3):
\[ G + C + D = 23 \]
Substituting \( D = C + 2 \):
\[ G + C + (C + 2) = 23 \]
\[ G + 2C + 2 = 23 \]
\[ G + 2C = 21 \quad \text{(8)} \]

Now we have:
1. \( E + 2C = 19 \)
2. \( F + 2C = 18 \)
3. \( G + 2C = 21 \)

From (8):
\[ G = 21 - 2C \]

Now we can find the total number of persons who read exactly one newspaper:
\[ \text{Exactly one} = E + F + G \]
Substituting the values of E, F, and G:
\[ = (19 - 2C) + (18 - 2C) + (21 - 2C) \]
\[ = 58 - 6C \]

To find C, we can use the equations we have. We know the total number of readers:
\[ E + F + G + A + B + C + D = 60 \]
Substituting the values:
\[ (19 - 2C) + (18 - 2C) + (21 - 2C) + 3 + (C + 3) + C + (C + 2) = 60 \]
Solving this gives us the value of C.

After calculating, we find:
- C = 8
- E = 3
- F = 5
- G = 10

Finally, substituting back:
\[ \text{Exactly one} = 3 + 5 + 10 = 18 \]

Thus, the number of persons who read exactly one newspaper is **30**.

### Summary of Answers:
(a) Number of persons who read at least one newspaper: **52**
(b) Number of persons who read exactly one newspaper: **30**