Practice Questions
Concept Explanation
When selections have restrictions, use complementary counting or split into valid cases.
Key Strategies
- Must include / must exclude: adjust r or use nCr on reduced set
- At least one: total − none
- Both roles filled from different pools: multiply independent choices
Example
Choose 3 from 7 with A mandatory: choose remaining 2 from other 6 → 6C2 = 15.
Common Trap
Subtract invalid cases from total rather than listing many valid cases when possible.
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Q1 Medium
Choose 3 from 7 with one specific item mandatory. Ways?
ExplanationPick mandatory item, then 6C2 = 15. -
Q2 Medium
Choose 2 from 8 with 2 specific items never together. Valid selections?
ExplanationTotal 8C2 = 28 minus 1 forbidden pair = 27. -
Q3 Medium
Choose at least 1 red from 4 red and 5 blue (pick 2 total). Ways?
ExplanationTotal 9C2 = 36 minus 5C2 = 10 (both blue) = 26. -
Q4 Medium
Choose 4 from 10 with 2 fixed members included. Ways?
ExplanationChoose 2 more from remaining 8: 8C2 = 28. -
Q5 Hard
Select 3 from 5 boys and 4 girls with at least 1 girl. Ways?
ExplanationTotal 9C3 = 84 minus 5C3 = 10 (no girls) = 74. -
Q6 Medium
Choose 2 from {1,2,3,4,5} with sum even. Ways?
ExplanationBoth odd: 3C2 = 3. Both even: 2C2 = 1. Total = 4. -
Q7 Hard
Pick 3 cards from 52 with at least 1 ace (4 aces). Ways?
Explanation52C3 − 48C3 = 22100 − 17296 = 4804.
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Q8 Medium
Choose 4 from 12 with 1 specific person excluded. Ways?
ExplanationChoose 4 from remaining 11: 11C4 = 495. -
Q9 Medium
Select 4 from 6 engineers and 4 designers with exactly 2 engineers. Ways?
Explanation6C2 × 4C2 = 15 × 8 = 120. -
Q10 Medium
Choose 2 from 10 with persons A and B not both selected. Ways?
Explanation10C2 − 1 = 45 − 1 = 44. -
Q11 Hard
Select 3 from 8 with both X and Y never together. Ways?
ExplanationTotal 8C3 = 56 minus 6C1 = 6 (X,Y plus one other) = 50. -
Q12 Hard
Committee of 3 from 5 engineers and 3 designers with at least 2 engineers. Ways?
Explanation3 engineers: 5C3 = 10. Two engineers + one designer: 5C2 × 3 = 30. Total = 40. -
Q13 Medium
Choose 4 from 11 with A included and B excluded. Ways?
ExplanationChoose 3 from remaining 9 (excluding B): 9C3 = 84. -
Q14 Medium
Select 2 from 7 with at least 1 prime from {2,3,5,7}. Ways?
ExplanationTotal 7C2 = 21 minus 3C2 = 3 (both from {1,4,6}) = 18.
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DS1 Easy
How many ways to choose 3 from 7 with one item mandatory?
Statement IOne specific item must be included.
Statement IIOrder of selection does not matter.
ExplanationStatement I alone: 6C2 = 15. -
DS2 Hard
How many selections have at least one girl?
Statement IThere are 5 boys and 4 girls.
Statement IIA team of 3 is chosen.
ExplanationNeed both: 9C3 − 5C3 = 84 − 10 = 74.
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DS3 Medium
How many pairs exclude both A and B together?
Statement I8 people are available.
Statement IITwo specific people cannot both be selected.
ExplanationStatement I alone: 8C2 − 1 = 27. -
DS4 Hard
How many 3-card hands contain at least one ace?
Statement IThe deck has 52 cards including 4 aces.
Statement IIHands of exactly 3 cards are drawn.
ExplanationNeed both: 52C3 − 48C3 = 4804.