To solve this problem, we use the principle of inclusion-exclusion for sets. The goal is to find the total number of people surveyed.

Definitions:

Let:

• $B$: the set of people who selected Bus.
• $T$: the set of people who selected Train.
• $A$: the set of people who selected Automobile.

Given:

• $|B| = 30$ (Bus)
• $|T| = 30$ (Train)
• $|A| = 100$ (Automobile)
• $|B \cap T| = 15$ (Bus and Train)
• $|B \cap A| = 15$ (Bus and Automobile)
• $|T \cap A| = 20$ (Train and Automobile)
• $|B \cap T \cap A| = 5$ (all three methods)

We aim to find the total number of people surveyed: $|B \cup T \cup A|$.

Inclusion-Exclusion Formula:

The formula for the union of three sets is: $|B \cup T \cup A| = |B| + |T| + |A| - |B \cap T| - |B \cap A| - |T \cap A| + |B \cap T \cap A|$

Substituting Values:

Substitute the given values into the formula: $|B \cup T \cup A| = 30 + 30 + 100 - 15 - 15 - 20 + 5$

Simplify:

$|B \cup T \cup A| = 160 - 50 + 5 = 115$

Conclusion:

The total number of people surveyed is 115.

Do you want further clarification or details about the steps?

Here are some related questions to expand your understanding:

1. How does the inclusion-exclusion principle apply to four or more sets?
2. Can you represent this problem with a Venn diagram?
3. What happens if some data is missing? How would you adjust?
4. Why is the intersection of three sets added back in the formula?
5. Can you calculate the total number of overlaps (people counted in multiple sets)?

Tip: When using inclusion-exclusion, carefully handle intersections to avoid undercounting or overcounting.