An executive education program surveyed 400 professional candidates regarding their enroll… — GRE General Test

An executive education program surveyed $400$ professional candidates regarding their enrollment in three specialized tracks: Artificial Intelligence ( $A$ ), Financial Technology ( $F$ ), and Sustainable Energy ( $S$ ).

The survey revealed the following data:
- $40$ candidates were enrolled in none of the three tracks.
- $210$ candidates were enrolled in Artificial Intelligence.
- $180$ candidates were enrolled in Financial Technology.
- $150$ candidates were enrolled in Sustainable Energy.
- $90$ candidates were enrolled in both Artificial Intelligence and Financial Technology.
- $70$ candidates were enrolled in both Financial Technology and Sustainable Energy.
- $60$ candidates were enrolled in both Artificial Intelligence and Sustainable Energy.

How many candidates were enrolled in exactly one of the three tracks?

A
$100$
B
$140$
C
$180$
$220$ Cevap
E
$260$

Cevap

The number of candidates enrolled in exactly one of the three tracks is

220

To find the number of candidates enrolled in exactly one track, we first apply the Principle of Inclusion-Exclusion to find the triple intersection

|A \cap F \cap S|

|A \cup F \cup S| = |A| + |F| + |S| - (|A \cap F| + |F \cap S| + |A \cap S|) + |A \cap F \cap S|

. Substituting known values gives

360 = 210 + 180 + 150 - 220 + |A \cap F \cap S|

, so

|A \cap F \cap S| = 40

. Isolating mutually exclusive regions yields

100

candidates in Artificial Intelligence only,

60

in Financial Technology only, and

60

in Sustainable Energy only. Summing these gives

100 + 60 + 60 = 220

Adım Adım Çözüm

Determine the number of candidates enrolled in at least one track.

|A \cup F \cup S| = 400 - 40 = 360

Candidates in none of the tracks must be subtracted from the total survey population.

Calculate the number of candidates enrolled in all three tracks (

x = |A \cap F \cap S|

) using the Principle of Inclusion-Exclusion.

360 = 210 + 180 + 150 - (90 + 70 + 60) + x \implies 360 = 320 + x \implies x = 40

The sum of individual sets overcounts pairwise intersections and undercounts the triple intersection.

Calculate the counts for regions representing enrollment in exactly two tracks.

A \cap F \text{ only} = 90 - 40 = 50

F \cap S \text{ only} = 70 - 40 = 30

A \cap S \text{ only} = 60 - 40 = 20

Subtracting the triple intersection from each pairwise intersection isolates candidates enrolled in precisely two tracks.

Calculate the counts for candidates enrolled in exactly one track and sum them.

A \text{ only} = 210 - (50 + 20 + 40) = 100

;

F \text{ only} = 180 - (50 + 30 + 40) = 60

;

S \text{ only} = 150 - (20 + 30 + 40) = 60

. Total

= 100 + 60 + 60 = 220

Subtracting all overlap regions from each set total yields the single-track enrollment.

Anahtar Kavram

Principle of Inclusion-Exclusion for Three Sets

Soru

Cevap

Adım Adım Çözüm

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