To find the number of adults who did not travel by any of these modes (plane, train, or bus), let’s define each group clearly and follow through with a systematic approach.

Step-by-Step Solution

1. Identify the Sets and Overlaps Given:

   • $P$: Adults who traveled by plane.
   • $T$: Adults who traveled by train.
   • $B$: Adults who traveled by bus.

   The information given translates to:

   • $|P \setminus (T \cup B)| = 70$: Traveled only by plane.
   • $|T \setminus (P \cup B)| = 75$: Traveled only by train.
   • $|B \setminus (P \cup T)| = 30$: Traveled only by bus.
   • $|P \cap B \setminus T| = 70$: Traveled by plane and bus, but not by train.
   • $|P \cap T \cap B| = 35$: Traveled by all three modes (plane, train, and bus).
   • $|P \cup T| = 225$: Traveled by plane or train.

2. Use the Inclusion-Exclusion Principle for $|P \cup T \cup B|$:

   The Inclusion-Exclusion Principle states: $|P \cup T \cup B| = |P| + |T| + |B| - |P \cap T| - |P \cap B| - |T \cap B| + |P \cap T \cap B|$

3. Express Given Quantities in Terms of $|P \cup T \cup B|$:

   We know that:

   • $|P \cup T| = 225$
   • $|B \setminus (P \cup T)| = 30$, which means these 30 people traveled only by bus.

   Therefore, we can calculate $|P \cup T \cup B|$ by adding those who traveled by plane or train to those who traveled only by bus: $|P \cup T \cup B| = |P \cup T| + |B \setminus (P \cup T)| = 225 + 30 = 255$

4. Calculate Those Who Did Not Travel by Any Mode:

   Since the total number of surveyed adults is 285, we subtract those who traveled by at least one mode: $285 - |P \cup T \cup B| = 285 - 255 = 30$

Conclusion

30 adults did not travel by any of these modes of transportation.

Would you like more details on the reasoning or steps used here?

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1. How would the answer change if we included other forms of transportation?
2. Can the Inclusion-Exclusion Principle help in finding subsets within a dataset?
3. What if there were overlapping data between plane-only and train-only travelers?
4. How is this method applicable to probability problems?
5. What are other uses of the Inclusion-Exclusion Principle in combinatorics?

Tip: The Inclusion-Exclusion Principle is widely useful for overlapping groups—it’s a key concept in combinatorics for calculating union sizes.