Combinations
From a group of 7 men and 4 women, a committee of 5 persons is to be formed. In how many ways can this be done if the committee must have at least 2 women?
Select the correct option:
Solution
301
- Identify Cases: Since there are 4 women available, 'at least 2' means 2, 3, or 4 women.
- Calculate Each Case:
- Case 1 (2W, 3M): C(4,2)×C(7,3)=6×35=210.
- Case 2 (3W, 2M): C(4,3)×C(7,2)=4×21=84.
- Case 3 (4W, 1M): C(4,4)×C(7,1)=1×7=7.
- Sum the Mutually Exclusive Events:
- Total = 210+84+7=301.
- Verification: Note that we cannot have 5 women because only 4 exist.
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About This Question
- Subject
- mathematics
- Chapter
- permutations and combinations
- Topic
- combinations
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
301
- Identify Cases: Since there are 4 women available, 'at least 2' means 2, 3, or 4 women.
- Calculate Each Case:
- Case 1 (2W, 3M): C(4,2)×C(7,3)=6×35=210.
- Case 2 (3W, 2M): C(4,3)×C(7,2)=4×21=84.
- Case 3 (4W, 1M): C(4,4)×C(7,1)=1×7=7.
- Sum the Mutually Exclusive Events:
- Total = 210+84+7=301.
- Verification: Note that we cannot have 5 women because only 4 exist.
This hard difficulty mathematics question is from the chapter permutations and combinations, covering the topic of combinations. It appeared in the 2025 exam.
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