A corporate training agency surveyed 180 professionals to determine which of three skill w… — GMAT

A corporate training agency surveyed $180$ professionals to determine which of three skill workshops they attended: Leadership ( $L$ ), Negotiation ( $N$ ), and Communication ( $C$ ). The survey revealed that $90$ professionals attended Leadership, $80$ attended Negotiation, and $70$ attended Communication. Additionally, $35$ attended both Leadership and Negotiation, $25$ attended both Negotiation and Communication, $40$ attended both Leadership and Communication, and $20$ attended none of the three workshops. How many of the surveyed professionals attended exactly one workshop?

Cevap: 100 professionals

Cevap

The number of professionals who attended exactly one workshop is 100.

By applying the principle of inclusion-exclusion for three overlapping sets, the number of professionals attending at least one workshop is

180 - 20 = 160

. Setting up the equation

160 = 90 + 80 + 70 - (35 + 25 + 40) + x

reveals that

x = 20

professionals attended all three workshops. Subtracting

20

from each pairwise intersection yields the counts for those attending exactly two workshops (

15

5

, and

20

, totaling

40

). Subtracting the exactly-two count (

40

) and the all-three count (

20

) from the total attending at least one (

160

) gives

100

professionals who attended exactly one workshop.

Adım Adım Çözüm

Determine the number of professionals in the union of all three sets

|L \cup N \cup C| = 180 - 20 = 160

The total number of professionals who attended at least one workshop is equal to the total surveyed minus those who attended none.

Apply the principle of inclusion-exclusion for three sets to find the intersection of all three workshops

160 = 90 + 80 + 70 - (35 + 25 + 40) + |L \cap N \cap C| \implies |L \cap N \cap C| = 20

Summing individual set counts double-counts pairwise intersections and triple-counts the triple intersection, so we adjust using the standard three-set formula.

Calculate the number of professionals who attended exactly two workshops

(35 - 20) + (25 - 20) + (40 - 20) = 15 + 5 + 20 = 40

Each pairwise intersection includes those who attended all three workshops; subtracting the triple intersection leaves those in exactly two sets.

Calculate the number of professionals who attended exactly one workshop

\text{Exactly one} = 160 - 40 - 20 = 100

Subtracting the number of professionals who attended exactly two workshops and all three workshops from the total attending at least one leaves those attending exactly one workshop.

Anahtar Kavram

Overlapping Sets (Three-Set Inclusion-Exclusion Principle)

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