Let's break down the problem and solve it step by step, starting with the given information:

Variables:

Let:

• $C$ represent those who said cost is important.
• $G$ represent those who said gas mileage is important.
• $S$ represent those who said safety is important.

The given information is:

1. $|C| = 84$ (84 said cost is important).
2. 15 said cost but not gas mileage: $|C \cap \overline{G}| = 15$.
3. $|S| = 72$ (72 said safety is important).
4. 48 said all three: $|C \cap G \cap S| = 48$.
5. 56 said cost and safety: $|C \cap S| = 56$.
6. 25 said gas mileage but not safety: $|G \cap \overline{S}| = 25$.
7. 20 said gas mileage but not cost: $|G \cap \overline{C}| = 20$.

Part a: Draw the Venn Diagram

We’ll use this information to fill out a Venn diagram for the three sets $C$, $G$, and $S$. To do this, we’ll work from the innermost intersection (the people who mentioned all three factors) outward.

1. $|C \cap G \cap S| = 48$, so there are 48 people who said cost, gas mileage, and safety are all important.
2. $|C \cap S| = 56$, but this includes the people who said all three factors are important. Therefore, the number of people who said cost and safety but not gas mileage is: $|C \cap S \cap \overline{G}| = 56 - 48 = 8.$
3. 15 people said cost but not gas mileage. This includes people who said cost and safety but not gas mileage. So the number of people who said cost but neither safety nor gas mileage is: $|C \cap \overline{G} \cap \overline{S}| = 15 - 8 = 7.$
4. $|G \cap \overline{S}| = 25$ people said gas mileage but not safety. This includes those who said gas mileage but neither cost nor safety, so we need to subtract those who said cost and gas mileage but not safety. The number of people who said cost and gas mileage but not safety is: $|C \cap G \cap \overline{S}| = 84 - 15 = 69 \quad \text{(this includes all cost and gas mileage people)}.$ Therefore, the number of people who said gas mileage and cost but not safety is ( G \cap S