In a corporate workshop of 80 executives, 52 executives attended the Leadership module and… — GMAT

In a corporate workshop of $80$ executives, $52$ executives attended the Leadership module and $40$ executives attended the Negotiation module. If $18$ executives attended neither module, how many executives attended both modules?

Answer: 30 executives

Answer

30 executives attended both modules.

Applying the formula

\text{Total} = \text{Group A} + \text{Group B} - \text{Both} + \text{Neither}

yields

80 = 52 + 40 - \text{Both} + 18

. Simplifying gives

80 = 110 - \text{Both}

, so the number of executives who attended both modules is

30

Step-by-Step Solution

Determine the number of executives who attended at least one of the two modules.

Executives attending at least one module =

80 - 18 = 62

Subtracting those who attended neither module from the total gives the union of the two sets.

Apply the inclusion-exclusion principle to find the intersection of the two sets.

\text{Both} = \text{Leadership} + \text{Negotiation} - \text{At least one} = 52 + 40 - 62 = 30

Summing the participants of each module double-counts those who attended both, so subtracting the number attending at least one module isolates the overlap.

Key Concept

Two-Set Overlapping Sets Formula (Inclusion-Exclusion Principle)

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