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).
-
Q1 Easy
Committee of 3 from 10 people. Ways?
Explanation10C3 = 120. -
Q2 Medium
Committee of 4 from 7 men and 5 women with 2 men and 2 women. Ways?
Explanation7C2 × 5C2 = 21 × 10 = 210. -
Q3 Medium
Choose chair and vice-chair from 8 members (distinct roles). Ways?
Explanation8P2 = 8 × 7 = 56. -
Q4 Hard
Committee of 5 from 12; then 1 president chosen from committee. Total ways?
Explanation12C5 × 5 = 792 × 5 = 3960. -
Q5 Medium
Committee of 3 from 6 men and 4 women with at least 1 woman. Ways?
Explanation10C3 − 6C3 = 120 − 20 = 100. -
Q6 Easy
Subcommittee of 2 from 9. Ways?
Explanation9C2 = 36. -
Q7 Medium
Committee of 4 from 5 doctors and 3 nurses with exactly 2 doctors. Ways?
Explanation5C2 × 3C2 = 10 × 3 = 30.
-
Q8 Medium
Panel of 3 speakers from 7 candidates with speaking order fixed. Ways?
Explanation7P3 = 210. -
Q9 Hard
Committee of 6 from 14. Ways?
Explanation14C6 = 3003. -
Q10 Medium
Choose 3 representatives from 4 regions with 2,3,4,5 candidates each (one per region). Ways?
Explanation2 × 3 × 4 × 5 = 120. -
Q11 Medium
Committee of 4 from 8 with A and B not both on committee. Ways?
Explanation8C4 − 6C2 = 70 − 4 = 66. -
Q12 Hard
Board of 5 from 9 with chair mandatory from {P,Q}. Ways?
ExplanationChoose chair in 2 ways, then 4 from remaining 8: 2 × 8C4 = 140. -
Q13 Medium
Committee of 3 from 11 with at least one senior (4 seniors). Ways?
Explanation11C3 − 7C3 = 165 − 35 = 130. -
Q14 Hard
Committee of 4 from 6 with president and secretary as distinct roles from committee. Ways?
ExplanationChoose 4: 6C4 = 15. Assign president and secretary: 4P2 = 12. Total = 180.
-
DS1 Easy
How many committees of 3 can be formed?
Statement IThere are 10 people.
Statement IIOrder within the committee does not matter.
ExplanationStatement I alone: 10C3 = 120. -
DS2 Medium
How many mixed committees with 2 men and 2 women?
Statement IThere are 7 men and 5 women.
Statement IIThe committee has 4 members.
ExplanationNeed both: 7C2 × 5C2 = 210.
-
DS3 Easy
How many ways to assign chair and vice-chair?
Statement IThere are 8 members.
Statement IIThe two roles are distinct.
ExplanationStatement I alone: 8P2 = 56. -
DS4 Medium
How many committees include at least one woman?
Statement IThere are 6 men and 4 women.
Statement IICommittee size is 3.
ExplanationNeed both: 10C3 − 6C3 = 100.