A committee with at least one woman — Aptitude Question
A committee with at least one woman: From 5 men and 4 women, how many committees of 3 can be formed that include at… Answer and worked solution on CodeKairo.
- Topic: Permutations & Combinations
- Section: Quantitative Aptitude
- Difficulty: Medium
- Time target: 60 seconds
Question
From 5 men and 4 women, how many committees of 3 can be formed that include at least one woman?
Options
- 70
- 74
- 80
- 84
Show the answer and the worked solution
Correct answer: B. 74
Worked solution
- All committees of 3 from 9 people: C(9,3) = 84.
- Committees with no woman (all men): C(5,3) = 10.
- At least one woman: 84 − 10 = 74.
Answer: 74.
The idea
'At least one' = total − 'none'. Counting the cases 1, 2, 3 women separately gives the same 40 + 30 + 4 = 74 but takes three times as long.
Shortcut. Count the single failing case instead of three good ones: C(5,3) = 10 all-male committees, so 84 − 10 = 74.
Common trap. 84 — every committee of 3 from 9, forgetting to remove the 10 made of men only.
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