In a survey of 100 persons it was found that 28 read magazine A, 30 read magazine B, 42 read magazine C, 8 read magazines A and B , 10 read magazines A and C, 5 read magazines B and C and 3 read all three magazines. Find :
(i) How many read none of three magazines ?
(ii) How many read magazine C only ?

Answer

To solve the problem step-by-step, we will use the principle of inclusion-exclusion to find the required values.

### Step 1: Define the Variables
Let:
- \( n(A) = 28 \) (number of people who read magazine A)
- \( n(B) = 30 \) (number of people who read magazine B)
- \( n(C) = 42 \) (number of people who read magazine C)
- \( n(A \cap B) = 8 \) (number of people who read both A and B)
- \( n(A \cap C) = 10 \) (number of people who read both A and C)
- \( n(B \cap C) = 5 \) (number of people who read both B and C)
- \( n(A \cap B \cap C) = 3 \) (number of people who read all three magazines)

### Step 2: Use Inclusion-Exclusion Principle
To find the total number of people who read at least one magazine, we use the formula:
\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) \]

Substituting the values:
\[ n(A \cup B \cup C) = 28 + 30 + 42 - 8 - 10 - 5 + 3 \]

### Step 3: Calculate the Total
Now, calculate the total:
\[ n(A \cup B \cup C) = 28 + 30 + 42 - 8 - 10 - 5 + 3 = 80 - 23 = 57 \]

### Step 4: Find the Number of People Who Read None of the Magazines
Since there are 100 persons surveyed:
\[ \text{Number of people who read none of the magazines} = 100 - n(A \cup B \cup C) = 100 - 57 = 43 \]

### Step 5: Find the Number of People Who Read Magazine C Only
To find the number of people who read only magazine C, we can use the formula:
\[ n(C \text{ only}) = n(C) - (n(A \cap C) + n(B \cap C) - n(A \cap B \cap C)) \]

Substituting the values:
\[ n(C \text{ only}) = 42 - (10 + 5 - 3) = 42 - 12 = 30 \]

### Final Answers
(i) The number of people who read none of the three magazines is **43**.

(ii) The number of people who read magazine C only is **30**.