In a town three newspapers A, B and C are published . 42% of the people in that town read A, 68% read B, 51% read C, 30% read A and B, 28% read B and C , 36% A and C and 18% do not read any paper . Find the % of population of town that reads all the three.

Options

A. 0.15
B. 0.25
C. 0.2
D. 0.35

Explanation

To solve the problem step by step, we will use the principle of inclusion-exclusion to find the percentage of the population that reads all three newspapers A, B, and C.

### Step 1: Understand the given data
We have the following percentages:
- \( n(A) = 42\% \) (people who read newspaper A)
- \( n(B) = 68\% \) (people who read newspaper B)
- \( n(C) = 51\% \) (people who read newspaper C)
- \( n(A \cap B) = 30\% \) (people who read both A and B)
- \( n(B \cap C) = 28\% \) (people who read both B and C)
- \( n(A \cap C) = 36\% \) (people who read both A and C)
- \( n(A \cup B \cup C) \) is what we need to find.
- \( 18\% \) do not read any newspaper.

### Step 2: Calculate the total percentage of people who read at least one newspaper
Since \( 18\% \) do not read any paper, the percentage of people who read at least one newspaper is:
\[ n(A \cup B \cup C) = 100\% - 18\% = 82\% \]

### Step 3: Apply the principle of inclusion-exclusion
Using the formula for the union of three sets:
\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C) \]
Let \( x = n(A \cap B \cap C) \), the percentage of people who read all three newspapers.

Substituting the known values:
\[ 82\% = 42\% + 68\% + 51\% - 30\% - 28\% - 36\% + x \]

### Step 4: Simplify the equation
Now, we will simplify the right side:
\[ 82\% = 42 + 68 + 51 - 30 - 28 - 36 + x \]
Calculating the sum:
\[ 42 + 68 + 51 = 161 \]
And the sum of the intersections:
\[ 30 + 28 + 36 = 94 \]
Now substituting these values back into the equation:
\[ 82 = 161 - 94 + x \]
\[ 82 = 67 + x \]

### Step 5: Solve for \( x \)
To find \( x \):
\[ x = 82 - 67 \]
\[ x = 15 \]

### Step 6: Conclusion
Thus, the percentage of the population that reads all three newspapers A, B, and C is:
\[ \boxed{15\%} \]