Practice Questions
Concept Explanation
Committee formation problems select groups with role or composition constraints. Order within a committee usually does not matter.
Key Strategies
- Chair + members: pick chair first, then combine remaining
- Gender/category splits: multiply combinations across categories
- At least/at most: use complementary counting
Example
Committee of 4 from 7 men and 5 women with 2 of each: 7C2 × 5C2 = 21 × 10 = 210.
Common Trap
Distinguish whether roles are assigned (ordered) or merely group membership (unordered).
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Q1 EasyA plain committee of 3 is to be formed from 10 people. In how many ways can this be done?ExplanationOrder within a committee does not matter: 10C3 = (10 × 9 × 8) / (3 × 2 × 1) = 120. Hence the answer is 120.
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Q2 EasyA subcommittee of 2 is chosen from 9 people. How many ways?Explanation9C2 = (9 × 8) / 2 = 36. Hence the answer is 36.
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Q3 MediumFrom 8 men and 5 women, a committee of 5 with exactly 3 men is to be formed. In how many ways can this be done?ExplanationChoose 3 men from 8 and 2 women from 5: 8C3 × 5C2 = 56 × 10 = 560. Hence the answer is 560.
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Q4 MediumA committee of 4 is formed from 7 men and 5 women with exactly 2 men and 2 women. How many such committees are possible?Explanation7C2 × 5C2 = 21 × 10 = 210. Hence the answer is 210.
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Q5 MediumA president and a secretary are to be chosen from 8 men and 6 women such that the president is a man and the secretary is a woman. In how many ways can this be done?ExplanationPresident: 8 choices (men). Secretary: 6 choices (women). Roles are distinct, so total = 8 × 6 = 48. Hence the answer is 48.
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Q6 MediumA committee of 4 from 5 doctors and 3 nurses must contain exactly 2 doctors. How many ways?Explanation5C2 × 3C2 = 10 × 3 = 30. Hence the answer is 30.
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Q7 MediumA committee of 4 is chosen from 8 people so that two particular persons are never together. In how many ways can this be done?ExplanationTotal committees = 8C4 = 70. Both particular persons included: choose 2 more from the remaining 6 → 6C2 = 15. Never together = 70 − 15 = 55. Hence the answer is 55.
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Q8 MediumIn how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?ExplanationRequired = 7C5 × 3C2 = 21 × 3 = 63. (Equivalently 7C2 × 3C1 = 21 × 3 = 63.) Hence the answer is 63.
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Q9 MediumA committee of 3 from 11 people must include at least one senior (there are 4 seniors). How many ways?ExplanationTotal = 11C3 = 165. No senior = 7C3 = 35. At least one senior = 165 − 35 = 130. Hence the answer is 130.
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Q10 HardFrom 12 members, a committee of 5 is formed and then one president is chosen from that committee. How many outcomes are possible?ExplanationFirst choose the committee: 12C5 = 792. Then choose president from the 5: 5 ways. Total = 792 × 5 = 3960. Hence the answer is 3960.
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Q11 HardIn how many ways can a committee consisting of 5 men and 6 women be formed from 8 men and 10 women?Explanation8C5 × 10C6 = 56 × 210 = 11760. Hence the answer is 11760.
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Q12 HardA committee of 5 is to be formed from 6 men and 4 women. In how many ways can this be done if two particular women refuse to serve together?ExplanationTotal committees = 10C5 = 252. Committees containing both particular women: choose the remaining 3 from the other 8 people → 8C3 = 56. Valid = 252 − 56 = 196. Hence the answer is 196.
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Q13 HardHow many ways can a board of 6 be chosen from 14 candidates?Explanation14C6 = 14! / (6! × 8!) = (14 × 13 × 12 × 11 × 10 × 9) / (6 × 5 × 4 × 3 × 2 × 1) = 3003. Hence the answer is 3003.
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Q14 HardFrom 6 people a committee of 4 is formed, and then a president and a secretary are appointed from the committee (distinct posts). How many ways?ExplanationChoose the committee: 6C4 = 15. Assign president and secretary from the 4: 4P2 = 4 × 3 = 12. Total = 15 × 12 = 180. Hence the answer is 180.
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DS1 Easy
How many committees of 3 can be formed from a group?
Statement IThere are 10 people in the group.
Statement IIOrder within the committee does not matter.
ExplanationStatement I with the standard unordered-committee rule gives 10C3 = 120. Statement II alone lacks n. Answer: A. -
DS2 Easy
In how many ways can a president (man) and secretary (woman) be chosen?
Statement IThere are 8 men and 6 women.
Statement IIThe president must be a man and the secretary a woman.
ExplanationStatement I alone does not force the gender–role rule. Statement II alone lacks the counts. Together: 8 × 6 = 48. Answer: C.
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DS3 Medium
How many committees have exactly 3 men and 2 women?
Statement IThere are 8 men and 5 women.
Statement IIThe committee has 5 members with exactly 3 men.
ExplanationStatement I alone lacks committee size/composition. Statement II alone lacks the sizes of the men and women pools. Together: 8C3 × 5C2 = 560. Answer: C. -
DS4 Hard
How many committees of 5 from 6 men and 4 women exclude a forbidden pair of women?
Statement ITwo particular women refuse to serve together.
Statement IICommittee size is 5 from the 10 people.
ExplanationTogether: 10C5 − 8C3 = 252 − 56 = 196. Answer: C.