A corporate venture capital firm evaluated 120 technology startups for investment. Each st… — GMAT

A corporate venture capital firm evaluated $120$ technology startups for investment. Each startup met at least one of three key criteria: strong artificial intelligence capability, established revenue growth, or international market presence. Exactly $65$ startups met the artificial intelligence criteria, $55$ met the revenue growth criteria, and $50$ met the international presence criteria. Furthermore, $25$ startups met both the artificial intelligence and revenue growth criteria, $20$ met both the revenue growth and international presence criteria, and $15$ met both the artificial intelligence and international presence criteria. How many startups met all three investment criteria?

A
$5$
$10$ Answer
C
$15$
D
$20$
E
$25$

Answer

The number of startups that met all three investment criteria is 10.

According to the Inclusion-Exclusion Principle for three sets, the total population is equal to the sum of the three individual sets minus the sum of the three pairwise intersections, plus the intersection of all three sets. Substituting the given values:

120 = 65 + 55 + 50 - (25 + 20 + 15) + x

, which simplifies to

120 = 110 + x

, giving

x = 10

. Thus,

10

startups met all three criteria.

Step-by-Step Solution

State the Principle of Inclusion-Exclusion for three sets

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

To account for startups counted in multiple overlapping categories without double-counting or over-subtracting.

Substitute the known values from the problem statement into the formula

120 = 65 + 55 + 50 - (25 + 20 + 15) + x

, where

x = |A \cap B \cap C|

Every startup meets at least one criterion, so the union size equals the total number of startups (

120

Simplify the sums and solve for

x

120 = 170 - 60 + x \implies 120 = 110 + x \implies x = 10

Subtracting

110

from

120

gives the exact number of startups meeting all three criteria.

Key Concept

Three-Set Inclusion-Exclusion Principle

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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