An urban planning department surveyed a total of 360 commercial buildings regarding three … — GRE General Test

An urban planning department surveyed a total of $360$ commercial buildings regarding three sustainability certifications: LEED ( $L$ ), BREEAM ( $B$ ), and WELL ( $W$ ). Exactly $50$ of the surveyed buildings held none of the three certifications. The survey revealed the following data:

• $180$ buildings held the LEED certification.
• $140$ buildings held the BREEAM certification.
• $130$ buildings held the WELL certification.
• $45$ buildings held both the LEED and BREEAM certifications.
• $40$ buildings held both the BREEAM and WELL certifications.
• $25$ buildings held all three certifications.

How many of the surveyed buildings held EXACTLY TWO of the three certifications?

Answer: 90 buildings

Answer

To find the number of buildings with exactly two certifications, we first determine the total number of buildings holding at least one certification:

360 - 50 = 310

. Applying the three-set inclusion-exclusion formula,

310 = 180 + 140 + 130 - (45 + 40 + |L \cap W|) + 25

, which simplifies to

310 = 390 - |L \cap W|

, giving

|L \cap W| = 80

. Next, we isolate the regions holding exactly two certifications by subtracting the 25 triple-certified buildings from each pairwise intersection: LEED & BREEAM only =

45 - 25 = 20

, BREEAM & WELL only =

40 - 25 = 15

, and LEED & WELL only =

80 - 25 = 55

. Summing these three exclusive regions gives

20 + 15 + 55 = 90

Step-by-Step Solution

Determine the total size of the union of all three sets

|L \cup B \cup W| = 360 - 50 = 310

Buildings holding at least one certification represent the entire surveyed population minus those holding no certifications.

Solve for the unknown overlap of LEED and WELL certifications using the inclusion-exclusion formula

|L \cap W| = 80

Using

|L \cup B \cup W| = |L| + |B| + |W| - |L \cap B| - |B \cap W| - |L \cap W| + |L \cap B \cap W|

, we have

310 = 180 + 140 + 130 - 45 - 40 - |L \cap W| + 25 = 390 - |L \cap W|

Calculate the number of buildings in each region corresponding to exactly two certifications

LEED and BREEAM only:

45 - 25 = 20

; BREEAM and WELL only:

40 - 25 = 15

; LEED and WELL only:

80 - 25 = 55

Each total pairwise intersection includes the 25 buildings that hold all three certifications, so subtracting 25 isolates those holding exclusively two certifications.

Sum the three isolated regions

20 + 15 + 55 = 90

The total number of buildings holding exactly two certifications is the sum of the three non-overlapping regions representing two certifications.

Key Concept

Three-Set Inclusion-Exclusion Principle and Partitioning Venn Diagrams

Estimated Time:2m 0s

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