UPSC Prelims 2007 Q142: Science & Technology – Quantitative Aptitude
Each of the 3 persons is to be given some identical items such that the product of the
numbers of items received by each of the three persons is equal to 30. In how many maximum
different ways can this distribution be done?
- (a) 21
- (b) 24
- (c) 27
- (d) 33
Answer: (c) 27
Explanation
Three people, product of items = 30
Each person must get at least 1 item, and the three numbers must multiply to 30.
Prime factors: 30 = 2 × 3 × 5 — three different primes.
Each prime can go to any one of the 3 people.
The choices are independent:
| Prime | Choices |
|---|---|
| 2 | 3 people |
| 3 | 3 people |
| 5 | 3 people |
| Total | 3 × 3 × 3 = 27 |
Check by listing types:
| Type | Example | Arrangements |
|---|---|---|
| (30, 1, 1) | one gets 30 | 3 |
| (a, b, 1) with ab = 30, a ≠ b | (2,15), (3,10), (5,6) | 3 × 6 = 18 |
| (2, 3, 5) | all different | 6 |
| Total | All three types | 27 |
Answer: (c) 27.
Why each option is right or wrong
- (a) Incorrect. 21 misses the arrangements of (2, 3, 5).
- (b) Incorrect. 24 misses the (30, 1, 1) cases.
- (c) CORRECT. Each of the 3 primes goes to one of 3 people: 3³ = 27.
- (d) Incorrect. 33 over-counts; each prime has 3 choices, so 27.
Key point
30 = 2 × 3 × 5; each prime goes to one of 3 people → 3³ = 27 ordered ways.
Remember
3 people, product 30 → 27 ways.
Common mistakes
- Counting only unordered splits — the persons are different, so order matters.
- Forgetting that someone can get 1 item (all primes to others).
- Trying to list everything without the prime-distribution shortcut.
Exam tip
Product = n with distinct primes p₁…pₖ, shared among r people → rᵏ ways.
Sources
- NCERT Class XI — Mathematics, Ch 6 (Permutations and Combinations)
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)