We can solve this using the principle of inclusion-exclusion for sets. Let's define the following:

• Let $P$ be the set of people who traveled by plane.
• Let $T$ be the set of people who traveled by train.
• Let $B$ be the set of people who traveled by bus.

We are given the following information:

• $|P \setminus (T \cup B)| = 65$, the number of people who traveled by plane but not by train or bus.
• $|T \setminus (P \cup B)| = 45$, the number of people who traveled by train but not by plane or bus.
• $|B \setminus (P \cup T)| = 35$, the number of people who traveled by bus but not by plane or train.
• $|P \cap B \setminus T| = 70$, the number of people who traveled by both plane and bus but not by train.
• $|P \cap T \cap B| = 45$, the number of people who traveled by all three modes.
• $|P \cup T| = 180$, the number of people who traveled by either plane or train.

We need to determine how many people did not travel by any mode of transportation.

Step 1: Total who traveled by plane and/or train

From the inclusion-exclusion principle, the number of people who traveled by plane or train is already given as $|P \cup T| = 180$.

Step 2: Total number who traveled by bus

To find how many people traveled by bus (whether or not they traveled by other modes), we need to consider:

• Those who traveled by bus only: $|B \setminus (P \cup T)| = 35$.
• Those who traveled by both plane and bus but not by train: $|P \cap B \setminus T| = 70$.
• Those who traveled by all three modes: $|P \cap T \cap B| = 45$.

So, the total number of people who traveled by bus is: $|B| = 35 + 70 + 45 = 150.$

Step 3: Combine with plane and train data

Since $|P \cup T| = 180$, and from this set, some also traveled by bus (specifically, those who traveled by all three modes or by bus and plane), the total number of people who traveled by any mode of transportation is: $|P \cup T \cup B| = |P \cup T| + |B| - \text{those who traveled by bus already counted in } P \cup T.$

But $|P \cup T| = 180$, and we've already counted:

• $70$ who traveled by both plane and bus but not by train,
• $45$ who traveled by all three modes.

So, we subtract these 115 people from the total of 150 people who traveled by bus: $|P \cup T \cup B| = 180 + (150 - 115) = 180 + 35 = 215.$

Step 4: People who traveled by none of the modes

The total number of people surveyed is 245. Therefore, the number of people who did not travel by any of these modes is: $245 - 215 = 30.$

Thus, 30 people did not travel by any of these modes of transportation.

Would you like further details or have any questions?

Here are some related questions you might explore:

1. How many people traveled by exactly two modes of transportation?
2. How many people traveled only by bus?
3. How many people traveled by either train or bus but not by plane?
4. What percentage of people traveled by at least one mode of transportation?
5. How many people traveled by train or bus but not by plane?

Tip: When dealing with multiple sets, diagrams like Venn diagrams can be helpful in visualizing intersections and unions of sets.