Understanding the Newspaper Survey Problem

This problem involves analyzing data from a survey where people read different newspapers. We are given the total number of people surveyed and the number of people reading specific combinations of three newspapers: H, I, and T. The goal is to find the number of people who read exactly one newspaper.

Given Information from the Survey

Let's represent the sets of people reading newspapers H, I, and T by $H$, $I$, and $T$ respectively. The given information is:

• Total number of people = 60
• Number of people who read newspaper H, $n(H) = 25$
• Number of people who read newspaper I, $n(I) = 26$
• Number of people who read newspaper T, $n(T) = 26$
• Number of people who read both H and I, $n(H \cap I) = 9$
• Number of people who read both H and T, $n(H \cap T) = 11$
• Number of people who read both T and I, $n(I \cap T) = 8$
• Number of people who read all three newspapers (H, I, and T), $n(H \cap I \cap T) = 3$

Finding Exactly One Newspaper Readers

We want to find the number of people who read only newspaper H, plus the number who read only newspaper I, plus the number who read only newspaper T. Using the principle of inclusion-exclusion, we can find the number of people reading exactly one newspaper.

Let's denote the number of people reading exactly one newspaper as $n(\text{exactly one})$. This can be calculated using the formula:

$\qquad n(\text{exactly one}) = n(H) + n(I) + n(T) - 2 \times [n(H \cap I) + n(H \cap T) + n(I \cap T)] + 3 \times n(H \cap I \cap T)$

Let's substitute the given values into the formula:

$\qquad n(\text{exactly one}) = 25 + 26 + 26 - 2 \times (9 + 11 + 8) + 3 \times 3$

First, sum the number of people reading individual newspapers:

$\qquad 25 + 26 + 26 = 77$

Next, sum the number of people reading exactly two newspapers (before subtracting the 'all three' overlap):

$\qquad 9 + 11 + 8 = 28$

Then, calculate twice this sum:

$\qquad 2 \times 28 = 56$

Calculate three times the number reading all three:

$\qquad 3 \times 3 = 9$

Now, substitute these results back into the formula for $n(\text{exactly one})$:

$\qquad n(\text{exactly one}) = 77 - 56 + 9$

$\qquad n(\text{exactly one}) = 21 + 9$

$\qquad n(\text{exactly one}) = 30$

Alternatively, we can find the number of people reading only one specific newspaper first:

• Only H: $n(H) - n(H \cap I) - n(H \cap T) + n(H \cap I \cap T) = 25 - 9 - 11 + 3 = 25 - 20 + 3 = 8$
• Only I: $n(I) - n(H \cap I) - n(I \cap T) + n(H \cap I \cap T) = 26 - 9 - 8 + 3 = 26 - 17 + 3 = 12$
• Only T: $n(T) - n(H \cap T) - n(I \cap T) + n(H \cap I \cap T) = 26 - 11 - 8 + 3 = 26 - 19 + 3 = 10$

The total number of people reading exactly one newspaper is the sum of these values:

$\qquad \text{Exactly one} = \text{Only H} + \text{Only I} + \text{Only T} = 8 + 12 + 10 = 30$

Both methods confirm that 30 people read exactly one newspaper.

Conclusion

Based on the survey data and using set theory principles, the number of people who read exactly one newspaper is 30.