A financial consulting firm audited 500 corporate investment portfolios for exposure to th… — GMAT

A financial consulting firm audited $500$ corporate investment portfolios for exposure to three alternative asset classes: Private Equity ( $P$ ), Venture Capital ( $V$ ), and Infrastructure ( $I$ ). Every audited portfolio contained at least one of the three asset classes.

The audit revealed the following:
- Exactly $270$ portfolios contained Private Equity.
- Exactly $250$ portfolios contained Venture Capital.
- Exactly $220$ portfolios contained Infrastructure.
- The ratio of the number of portfolios containing both Private Equity and Venture Capital to those containing both Venture Capital and Infrastructure to those containing both Private Equity and Infrastructure was $11 : 9 : 10$ , respectively.
- The number of portfolios containing all three asset classes was $60$ .

How many portfolios contained exactly two of the three asset classes?

Answer: 120 portfolios

Answer

The total number of portfolios containing exactly two of the three asset classes is 120.

Using the 3-set inclusion-exclusion formula, we establish that

500 = 270 + 250 + 220 - (11k + 9k + 10k) + 60

, which simplifies to

30k = 300

, so

k = 10

. The sum of the pairwise intersections is

110 + 90 + 100 = 300

. Since each pairwise intersection includes the 60 portfolios that contain all three asset classes, the number of portfolios containing exactly two asset classes is

(110-60) + (90-60) + (100-60) = 50 + 30 + 40 = 120

Step-by-Step Solution

Set up the 3-Set Inclusion-Exclusion Principle equation

\text{Total} = |P| + |V| + |I| - (|P \cap V| + |V \cap I| + |P \cap I|) + |P \cap V \cap I|

This formula accounts for all region overlaps without double-counting or triple-counting.

Define variables using the given ratio for pairwise intersections

|P \cap V| = 11k

|V \cap I| = 9k

, and

|P \cap I| = 10k

Expressing the pairwise intersection sizes with a common ratio constant

k

allows us to substitute them into a single-variable linear equation.

Substitute the known numerical values and solve for

k

500 = 270 + 250 + 220 - (11k + 9k + 10k) + 60 \implies 500 = 800 - 30k \implies k = 10

Using algebraic simplification to solve for

k

gives the scaling factor for the pairwise intersections.

Calculate the actual sizes of the pairwise intersections

|P \cap V| = 110

|V \cap I| = 90

, and

|P \cap I| = 100

; Sum of pairwise intersections =

300

Multiplying each ratio term by

k=10

gives the exact count of portfolios containing at least the respective pairs.

Calculate portfolios containing exactly two asset classes

(110 - 60) + (90 - 60) + (100 - 60) = 50 + 30 + 40 = 120

The triple intersection (

60

) must be subtracted from each pairwise overlap to isolate the regions corresponding to portfolios holding exactly two asset classes.

Key Concept

Three-Set Overlapping Venn Diagrams and Inclusion-Exclusion Principle

Estimated Time:2m 0s

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