From a group of 7 men and 6 women, a 5-person committee is to be formed such that at least 3 men are included. In how many ways can this be done?

Problem restatementSelect 5 people from 7 men and 6 women with at least 3 men on the committee.

Given data

• Men = 7, Women = 6
• Committee size = 5
• At least 3 men

Concept/ApproachCasework on the number of men: 3 men + 2 women; 4 men + 1 woman; 5 men + 0 women.

Step-by-Step calculationCase (3M,2W): ⁡7C3⁡6C2 = 35 × 15 = 525Case (4M,1W): ⁡7C4⁡6C1 = 35 × 6 = 210Case (5M,0W): ⁡7C5 = 21Total ways = 525 + 210 + 21 = 756

Common pitfallsIncluding the (2M,3W) case (violates the constraint) or using permutations instead of combinations.

Final Answer756