From a group of 7 men and 6 women, in how many ways can a committee of 5 persons be formed such that at least 3 members of the committee are men?

Introduction / Context:This question is a standard committee selection problem involving combinations and a restriction on the minimum number of men. Such problems frequently appear in aptitude and competitive exams to test your ability to handle multiple cases and combine them correctly.

Given Data / Assumptions:

• Total men available: 7.
• Total women available: 6.
• Committee size: 5 persons.
• Condition: At least 3 members must be men.
• Order of selection does not matter; only the combination of people matters.

Concept / Approach:“At least 3 men” means the committee can have 3 men and 2 women, 4 men and 1 woman, or 5 men and 0 women. We calculate the number of committees for each case using combinations and then add the results. The general combination formula is nCr = n! / (r! * (n - r)!).

Step-by-Step Solution:Case 1: 3 men and 2 women.Number of ways to choose men: 7C3 = 35.Number of ways to choose women: 6C2 = 15.Total for this case: 35 * 15 = 525.Case 2: 4 men and 1 woman.Ways to choose men: 7C4 = 35.Ways to choose women: 6C1 = 6.Total for this case: 35 * 6 = 210.Case 3: 5 men and 0 women.Ways to choose men: 7C5 = 21.Total for this case: 21.Now add all cases: 525 + 210 + 21 = 756.

Verification / Alternative check:Total possible 5 person committees from 13 people is 13C5.You could compute committees with 0, 1 or 2 men and subtract from 13C5, but the direct case wise method above is simpler and less error prone here.

Why Other Options Are Wrong:564, 735 and 657 arise from either omitting one of the required cases or miscomputing one of the combinations.None of these match the correct total of 756.

Common Pitfalls:Misinterpreting “at least 3 men” as “exactly 3 men”.Forgetting to include the all men case (5 men and 0 women).Using permutations instead of combinations, which would incorrectly treat different orders as distinct committees.

Final Answer:The number of ways to form such a committee is 756.