UPSC Prelims 2006 Q68: Science & Technology – Quantitative Aptitude
3 digits are chosen at random from 1, 2, 3, 4, 5, 6, 7, 8 and 9 without repeating any digit.
What is the probability that their product is odd?
- (a) 2/3
- (b) 5/108
- (c) 5/42
- (d) 7/48
Answer: (c) 5/42
Explanation
Probability that the product of three chosen digits is odd
The product is odd only when all three digits are odd.
| Step | Calculation |
|---|---|
| Odd digits among 1 to 9 | 1, 3, 5, 7, 9 = 5 |
| Ways to choose 3 odd digits | C(5, 3) = 10 |
| Ways to choose any 3 of 9 digits | C(9, 3) = 84 |
| Probability | 10 / 84 = 5/42 |
Answer: (c) 5/42.
Why each option is right or wrong
- (a) Incorrect. 2/3 is far larger than the true probability of all three digits being odd.
- (b) Incorrect. 5/108 does not match 10/84.
- (c) CORRECT. C(5,3)/C(9,3) = 10/84 = 5/42.
- (d) Incorrect. 7/48 does not match 10/84.
Key point
The product is odd only if all three digits are odd: C(5,3)/C(9,3) = 10/84 = 5/42.
Remember
Odd product from 3 distinct digits of 1-9: 5/42.
Common mistakes
- Using probabilities with replacement such as (5/9) cubed.
- Forgetting that a single even digit makes the product even.
- Not simplifying the fraction 10/84 to 5/42.
Exam tip
A product is odd only when every factor is odd.
Sources
- NCERT Class XI Mathematics — Probability
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Probability)