An automotive testing center evaluated 350 electric vehicle models for three advanced driv… — GMAT

An automotive testing center evaluated $350$ electric vehicle models for three advanced driver-assistance features: Lane Keeping Assist ( $L$ ), Automatic Emergency Braking ( $A$ ), and Blind Spot Detection ( $B$ ). The evaluation showed that $180$ models had feature $L$ , $150$ had feature $A$ , and $160$ had feature $B$ . Additionally, $65$ models had both $L$ and $A$ , $50$ had both $A$ and $B$ , $60$ had both $L$ and $B$ , and $25$ models had all three features. How many of the evaluated electric vehicle models had none of these three features?

Answer: 10 models

Answer

The number of electric vehicle models that had none of the three features is 10.

Using the formula for three overlapping sets,

|L \cup A \cup B| = |L| + |A| + |B| - (|L \cap A| + |A \cap B| + |L \cap B|) + |L \cap A \cap B|

, we get

180 + 150 + 160 - (65 + 50 + 60) + 25 = 340

vehicles with at least one feature. Subtracting this from the total evaluated group of

350

yields

350 - 340 = 10

vehicles with none of the features.

Step-by-Step Solution

Use the Principle of Inclusion-Exclusion for three overlapping sets to find the total number of models with at least one feature.

|L \cup A \cup B| = 180 + 150 + 160 - (65 + 50 + 60) + 25 = 340

models.

Adding individual set counts overcounts pairwise overlap regions twice and the central triple overlap three times. Subtracting pairwise intersections corrects for double counting, and adding back the triple intersection accounts for its over-subtraction.

Subtract the number of models having at least one feature from the total number of tested models.

\text{None} = 350 - 340 = 10

models.

The entire group consists of models with at least one feature plus models with none of the features.

Key Concept

Three-Set Inclusion-Exclusion Principle and Venn Diagram Region Partitioning

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