
In a street of 150 houses, three different newspapers are delivered:  the Tribune, the Herald, and the Chronicle, as follows:


• 40 receive the Tribune
• 35 receive the Herald
• 60 receive the Chronicle
• 7 receive the Tribune and the Herald
• 10 receive the Herald and the Chronicle
• 4 receive the Tribune and the Chronicle
• 34 receive no newspaper at all.


**How many receive all three newspapers?**


Note: If 7 receive the Tribune and the Herald’, this information does not mean that 7 people receive the Tribune and the Herald *only*.

Communicate all the working.
in this question the sum of all the value sums up to 190 whereas the value is supposed to be 150

This problem involves finding how many houses receive all three newspapers. The data given can be interpreted using set theory, and we will solve it step-by-step using a Venn diagram or inclusion-exclusion principle.

### Given Information:
Let:
- \( T \) represent the set of houses receiving the **Tribune**.
- \( H \) represent the set of houses receiving the **Herald**.
- \( C \) represent the set of houses receiving the **Chronicle**.

From the problem, we have the following:
- \( |T| = 40 \) (Houses receiving the Tribune)
- \( |H| = 35 \) (Houses receiving the Herald)
- \( |C| = 60 \) (Houses receiving the Chronicle)
- \( |T \cap H| = 7 \) (Houses receiving the Tribune and the Herald)
- \( |H \cap C| = 10 \) (Houses receiving the Herald and the Chronicle)
- \( |T \cap C| = 4 \) (Houses receiving the Tribune and the Chronicle)
- 34 houses receive no newspapers at all.

We are tasked with finding the number of houses that receive all three newspapers, denoted as \( x \).

### Step-by-Step Solution:

1. **Total number of houses:**
   The total number of houses is given as 150. Since 34 houses receive no newspapers, the number of houses receiving at least one newspaper is:
   \[
   150 - 34 = 116
   \]
   This means that 116 houses receive at least one newspaper.

2. **Inclusion-Exclusion Formula:**
   Using the inclusion-exclusion principle to calculate the total number of houses receiving at least one newspaper:
   \[
   |T \cup H \cup C| = |T| + |H| + |C| - |T \cap H| - |H \cap C| - |T \cap C| + |T \cap H \cap C|
   \]
   Plugging in the known values:
   \[
   116 = 40 + 35 + 60 - 7 - 10 - 4 + x
   \]
   Simplifying:
   \[
   116 = 135 - 21 + x
   \]
   \[
   116 = 114 + x
   \]
   Solving for \( x \):
   \[
   x = 116 - 114 = 2
   \]

### Answer:
Thus, the number of houses that receive all three newspapers is \( \boxed{2} \).

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Do you need further details or have any questions?

Here are 5 related questions you could explore:
1. How would the answer change if 5 houses received all three newspapers?
2. Can you calculate how many houses receive exactly two newspapers?
3. How many houses receive only the Tribune?
4. What is the total number of houses receiving only the Herald?
5. How many houses receive exactly one newspaper?

**Tip:** For problems involving overlapping groups, the inclusion-exclusion principle is a powerful tool for counting intersections.

In a street of 150 houses, three different newspapers are delivered: the Tribune, the Herald, and the Chronicle, as follows: • 40 receive the Tribune • 35 receive the Herald • 60 receive the Chronicle • 7 receive the Tribune and the Herald • 10 receive the Herald and the Chronicle • 4 receive the Tribune and the Chronicle • 34 receive no newspaper at all. How many receive all three newspapers? Note: If 7 receive the Tribune and the Herald’, this information does not mean that 7 people receive the Tribune and the Herald only.

Solve the problem of determining how many houses receive all three newspapers using set theory and the inclusion-exclusion principle. This problem involves the delivery of the Tribune, Herald, and Chronicle to 150 houses, and is perfect for students in Grades 9-12.

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