A committee of 5 members is to be formed from a pool of 6 men and 5 women. The committee m… — GMAT

A committee of $5$ members is to be formed from a pool of $6$ men and $5$ women. The committee must include at least $2$ men and at least $2$ women. However, two specific individuals in the pool, one man and one woman, refuse to serve on the committee together. How many different valid $5$ -member committees can be formed?

Answer: 280

Answer

The correct answer is 280.

The correct answer is 280, found by taking the total possible committees satisfying the gender constraint (350) and subtracting the invalid committees containing both conflicting members (70).

Step-by-Step Solution

Calculate total combinations satisfying gender constraints without individual restriction

350 valid gender-balanced committees (200 from 3M/2W and 150 from 2M/3W)

Establishes the total baseline number of committee selections before removing forbidden pairings

Calculate forbidden combinations containing both specific conflicting individuals

70 forbidden combinations (40 with 3M/2W overall and 30 with 2M/3W overall)

Identifies committee selections that violate the condition that the two individuals cannot serve together

Subtract forbidden combinations from total baseline combinations

280 valid committee selections

Applying complementary counting (

350 - 70

) yields the exact number of allowable committees

Key Concept

Combinations with group constraints and complementary counting

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