Of the 74 people who participated in a survey

12 enjoyed math and history,
16 did not enjoy science,
23 enjoyed history,
16 enjoyed science and history,
67 did not enjoy all three (math and science and history),
25 enjoyed math and science,
and 41 did not enjoy math. How many people who participated in the survey enjoyed at most one of the three options?

We are tasked with finding how many people enjoyed at most one of the three options: Math, Science, or History. The total number of survey participants is 74.

### Defining the variables:
- Let \( M \) represent the set of people who enjoyed Math.
- Let \( S \) represent the set of people who enjoyed Science.
- Let \( H \) represent the set of people who enjoyed History.

The data provided is as follows:
1. 12 people enjoyed both Math and History: \( |M \cap H| = 12 \)
2. 16 people did not enjoy Science: \( |S^c| = 16 \), which implies \( |S| = 74 - 16 = 58 \)
3. 23 people enjoyed History: \( |H| = 23 \)
4. 16 people enjoyed both Science and History: \( |S \cap H| = 16 \)
5. 67 people did not enjoy all three subjects: \( |(M \cap S \cap H)^c| = 67 \), implying that 7 people enjoyed all three subjects: \( |M \cap S \cap H| = 7 \)
6. 25 people enjoyed both Math and Science: \( |M \cap S| = 25 \)
7. 41 people did not enjoy Math: \( |M^c| = 41 \), implying \( |M| = 74 - 41 = 33 \)

### Step-by-Step Solution:

We will use the principle of inclusion-exclusion to determine how many people enjoyed at least one of the three subjects. This can then help us figure out how many enjoyed at most one subject.

1. **Apply inclusion-exclusion** to find the total number of people who enjoyed at least one of the three subjects:

\[
|M \cup S \cup H| = |M| + |S| + |H| - |M \cap S| - |S \cap H| - |M \cap H| + |M \cap S \cap H|
\]

Substituting the values:

\[
|M \cup S \cup H| = 33 + 58 + 23 - 25 - 16 - 12 + 7 = 68
\]

Thus, 68 people enjoyed at least one of the subjects.

2. **Find the number of people who enjoyed no subjects**:
\[
74 - 68 = 6
\]

So, 6 people did not enjoy any of the three subjects.

3. **Find the number of people who enjoyed exactly one subject**:

- To find how many people enjoyed only Math, we subtract those who enjoyed both Math and Science, Math and History, and all three from the total who enjoyed Math:
\[
|M \text{ only}| = |M| - |M \cap S| - |M \cap H| + |M \cap S \cap H|
\]
\[
|M \text{ only}| = 33 - 25 - 12 + 7 = 3
\]

- To find how many people enjoyed only Science, we subtract those who enjoyed both Science and Math, Science and History, and all three from the total who enjoyed Science:
\[
|S \text{ only}| = |S| - |M \cap S| - |S \cap H| + |M \cap S \cap H|
\]
\[
|S \text{ only}| = 58 - 25 - 16 + 7 = 24
\]

- To find how many people enjoyed only History, we subtract those who enjoyed both History and Math, History and Science, and all three from the total who enjoyed History:
\[
|H \text{ only}| = |H| - |M \cap H| - |S \cap H| + |M \cap S \cap H|
\]
\[
|H \text{ only}| = 23 - 12 - 16 + 7 = 2
\]

4. **Total number of people who enjoyed exactly one subject**:
\[
|M \text{ only}| + |S \text{ only}| + |H \text{ only}| = 3 + 24 + 2 = 29
\]

5. **Total number of people who enjoyed at most one subject**:

This includes people who enjoyed exactly one subject and those who enjoyed none:
\[
\text{At most one} = 29 + 6 = 35
\]

Thus, **35 people** enjoyed at most one of the three options (Math, Science, or History).

Would you like any further explanation or details?

### Follow-up questions:
1. How many people enjoyed exactly two of the subjects?
2. What is the probability that a randomly selected person enjoyed at least one subject?
3. How many people enjoyed Science but not Math?
4. Can you derive the formula used in inclusion-exclusion?
5. How many people enjoyed all three subjects, Math, Science, and History?

**Tip**: Use Venn diagrams to visualize problems involving sets and intersections for clearer understanding.

Of the 74 people who participated in a survey: 12 enjoyed math and history, 16 did not enjoy science, 23 enjoyed history, 16 enjoyed science and history, 67 did not enjoy all three (math and science and history), 25 enjoyed math and science, and 41 did not enjoy math. How many people who participated in the survey enjoyed at most one of the three options?

Analyze a survey where 74 people shared their preferences for math, science, and history. This problem uses the Inclusion-Exclusion Principle to find how many enjoyed at most one of the three subjects. Learn the key concepts in set theory and data analysis, suitable for Grades 10-12.

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