To analyze the data, let's break it into a Venn diagram problem to find out how many students are taking specific combinations of courses.

We are given the following data:

• Total students surveyed: $n = 86$
• $|M| = 35$ (students taking Math)
• $|E| = 63$ (students taking English)
• $|H| = 45$ (students taking History)
• $|M \cap E| = 29$ (students taking Math and English)
• $|M \cap H| = 16$ (students taking Math and History)
• $|E \cap H| = 32$ (students taking English and History)
• $|M \cap E \cap H| = 12$ (students taking all three)

Step 1: Use Inclusion-Exclusion Principle

We calculate the number of students in each region of the Venn diagram.

1. Students taking only Math and English, but not History: $|M \cap E| - |M \cap E \cap H| = 29 - 12 = 17$

2. Students taking only Math and History, but not English: $|M \cap H| - |M \cap E \cap H| = 16 - 12 = 4$

3. Students taking only English and History, but not Math: $|E \cap H| - |M \cap E \cap H| = 32 - 12 = 20$

4. Students taking all three courses: $|M \cap E \cap H| = 12$

5. Students taking only Math: Subtract students in overlapping regions from $|M|$: $|M| - (|M \cap E| + |M \cap H| - |M \cap E \cap H|) = 35 - (29 + 16 - 12) = 2$

6. Students taking only English: Subtract students in overlapping regions from $|E|$: $|E| - (|M \cap E| + |E \cap H| - |M \cap E \cap H|) = 63 - (29 + 32 - 12) = 14$

7. Students taking only History: Subtract students in overlapping regions from $|H|$: $|H| - (|M \cap H| + |E \cap H| - |M \cap E \cap H|) = 45 - (16 + 32 - 12) = 9$

8. Students not taking any of the three courses: Subtract students taking at least one course from the total: $n - (|M| + |E| + |H| - |M \cap E| - |M \cap H| - |E \cap H| + |M \cap E \cap H|) = 86 - 84 = 2$

Step 2: Summarize Results

Now, summarize the number of students in each region of the Venn diagram:

• Only Math: 2
• Only English: 14
• Only History: 9
• Math and English (only): 17
• Math and History (only): 4
• English and History (only): 20
• All three: 12
• None: 2

Would you like to see a visualization of this Venn diagram or have any part clarified?

5 Related Questions:

1. How can the Inclusion-Exclusion principle be extended for more than three sets?
2. How do you calculate probabilities for overlapping events in this context?
3. What would change if additional data, such as students taking none, were not given?
4. How would the solution change if more courses were added?
5. Can this data be represented using matrices or other mathematical structures?

Tip:

When solving Venn diagram problems, always work from the innermost overlaps (all sets) outward to avoid double-counting.