Practice Questions
Concept Explanation
Mixed P and C problems combine permutations, combinations, circular arrangements, and restrictions in multi-step placement scenarios.
Key Formulas
- nPr = n!/(n − r)!, nCr = n!/(r!(n − r)!)
- Circular: (n − 1)!; identical items: n!/(p!q!)
- Break problem into stages and multiply independent counts
Example
Choose 3 from 8, then arrange chosen 3 in a row: 8C3 × 3! = 8P3 = 336.
Common Trap
Decide at each step whether order matters before choosing nPr or nCr.
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Q1 Medium
Choose 3 from 8 and arrange them in a row. Total ways?
Explanation8C3 × 3! = 56 × 6 = 336 = 8P3. -
Q2 Medium
5 people around a table; then 1 chairperson chosen from them. Ways?
Explanation(5−1)! × 5 = 24 × 5 = 120. -
Q3 Hard
Arrange 3 from 7 and select 2 leaders from arranged group (ordered). Ways?
Explanation7P3 × 3P2 = 210 × 6 = 1260. -
Q4 Hard
Choose committee of 3 from 6 men and 4 women with more men than women. Ways?
Explanation3 men: 6C3 = 20. 2 men 1 woman: 6C2 × 4 = 60. Total = 80. -
Q5 Hard
Arrangements of MATHEMATICS with M at both ends:
ExplanationFix 2 M at ends; arrange remaining 9 letters with repeats: 9!/(2!×2!×2!) = 10080. -
Q6 Hard
Select 4 from 10 and arrange in a circle. Ways?
Explanation10C4 × 3! = 210 × 6 = 1260. -
Q7 Medium
3-digit numbers from {1,2,3,4,5} without repetition and ascending order automatically:
ExplanationChoose 3 digits from 5: 5C3 = 10.
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Q8 Easy
Arrange CAT and DOG in a row (each word kept intact). Ways?
Explanation2 blocks: 2! = 2. -
Q9 Medium
Choose 2 from 5 and arrange; choose 2 from remaining 3 and arrange. Total ways?
Explanation5P2 × 3P2 = 20 × 6 = 120. -
Q10 Hard
5 distinct beads on a necklace with flip symmetry. Arrangements?
Explanation(5−1)!/2 = 24/2 = 12. -
Q11 Medium
8C3 + 8P3 equals:
Explanation56 + 336 = 392. -
Q12 Medium
Ways to distribute 3 distinct prizes to 10 students (one prize each)?
Explanation10P3 = 720. -
Q13 Hard
Choose 5 from 12 and arrange in a row. Total ways?
Explanation12P5 = 95040. -
Q14 Medium
Arrangements of LETTER with L at start and R at end:
ExplanationFix L,R. Arrange E,T,E in middle: 3!/2! = 3.
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DS1 Easy
How many ways to choose and arrange 3 from 8 in a row?
Statement I8 distinct objects are available.
Statement II3 are chosen and arranged in a row.
ExplanationNeed both: 8P3 = 336. -
DS2 Medium
How many circular seating arrangements with a chairperson?
Statement I5 people sit around a round table.
Statement IIOne chairperson is chosen after seating.
ExplanationNeed both: 4! × 5 = 120.
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DS3 Medium
How many ascending 3-digit numbers from {1,2,3,4,5} without repetition?
Statement IDigits cannot repeat.
Statement IIDigits must appear in ascending order.
ExplanationStatement II alone: choose any 3 digits in one order = 5C3 = 10. -
DS4 Easy
What is 8C3 + 8P3?
Statement I8C3 = 56.
Statement II8P3 = 336.
ExplanationNeed both: 56 + 336 = 392.