Let's solve this step-by-step as you requested: --- ### **STEP 1: Introduction to the Concept** This problem involves understanding set operations with overlapping groups. We are given a survey of 22 vehicles and how many have features like audio systems, audio adaptors, and pay telephones. We need to find the number of vehicles that fit specific combinations of these features using the principle of inclusion and exclusion. --- ### **STEP 2: Read the Given Information and Assign Symbols** Let: - \( A \): Vehicles with audio window and audio adaptor = **15** - \( B \): Vehicles with pay telephones = **10** - \( A \cap B \): Vehicles with both audio window/audio adaptor and pay telephone = **5** - \( A \cup B \): Vehicles with at least one of these features = **15 + 10 - 5 = 20** --- ### **STEP 3: Answer Each Part** #### **a) Audio window and audio adaptor, but no pay telephone** This is \( A - (A \cap B) \): \[ 15 - 5 = 10 \] #### **b) Pay telephone** Already given as **10**. #### **c) Both audio window and pay telephone** Already given as **5**. #### **d) Audio window and audio adaptor, or pay telephone** This is \( A \cup B \): \[ 15 + 10 - 5 = 20 \] #### **e) Audio window and audio adaptor, but no pay telephone** This is the same as (a), already calculated: **10**. --- ### **STEP 4: Final Answers in Few Sentences** There are 10 vehicles with an audio window and audio adaptor but no pay telephone, 10 with a pay telephone, 5 with both features, and 20 with either or both features. --- ### **STEP 5: What We Did in Each Step** 1. Introduced the concept of set operations. 2. Assigned the information to set symbols. 3. Used set operations to answer each part. 4. Summarized the answers briefly. 5. Explained the steps taken concisely.