At an investment firm with 200 portfolio managers, 110 managers invest in Equities, 95 inv… — GMAT

At an investment firm with $200$ portfolio managers, $110$ managers invest in Equities, $95$ invest in Fixed Income, and $80$ invest in Commodities. Furthermore, $45$ managers invest in both Equities and Fixed Income, $35$ invest in both Equities and Commodities, and $30$ invest in both Fixed Income and Commodities. If $15$ managers do not invest in any of these three asset classes, how many portfolio managers invest in all three asset classes?

Cevap: 10 managers

Cevap

The correct answer is 10 portfolio managers.

Subtracting the 15 managers who do not invest in any asset class from the total 200 yields 185 managers in at least one asset class. Applying the 3-set inclusion-exclusion formula

185 = (110 + 95 + 80) - (45 + 35 + 30) + x

simplifies to

185 = 175 + x

, which gives

x = 10

for the number of managers investing in all three asset classes.

Adım Adım Çözüm

Determine the total number of portfolio managers investing in at least one asset class.

185 managers

Subtracting the 15 managers who invest in none of the three asset classes from the total of 200 gives the size of the union of the three sets:

200 - 15 = 185

Set up the Inclusion-Exclusion formula for three overlapping sets.

|E \cup F \cup C| = |E| + |F| + |C| - (|E \cap F| + |E \cap C| + |F \cap C|) + |E \cap F \cap C|

To avoid overcounting elements present in multiple set intersections, pairwise intersections are subtracted and the triple intersection is added back.

Substitute known values into the equation and solve for the unknown triple intersection

x

x = 10

185 = 110 + 95 + 80 - (45 + 35 + 30) + x \implies 185 = 285 - 110 + x \implies 185 = 175 + x \implies x = 10

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle

Soru

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Adım Adım Çözüm

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