UPSC Prelims 2008 Q148: Science & Technology – Quantitative Aptitude
There are 6 different letters and 6 correspondingly addressed envelopes. If the letters are
randomly put in the envelopes, what is the probability that exactly 5 letters go into the correctly
addressed envelopes?
- (a) Zero
- (b) 1/6
- (c) 1/2
- (d) 5/6
Answer: (a) Zero
Explanation
Probability that exactly 5 of 6 letters go into their correct envelopes
This is a classic probability puzzle revealing a surprising logical constraint.
| Point | Reasoning |
|---|---|
| Setup | 6 letters, 6 correspondingly addressed envelopes, placed randomly |
| Key insight | If exactly 5 letters are placed in their correct envelopes, the 6th letter has only one envelope left — and since all other envelopes are already correctly filled, that last envelope must also be the correct one for the last letter |
| Conclusion | It is logically impossible to have exactly 5 correct and 1 incorrect — if 5 are correct, the 6th must also be correct |
Why the probability is exactly zero:
| Point | Detail |
|---|---|
| If 5 are correctly placed | The last remaining letter and envelope must match by elimination |
| Therefore | 'Exactly 5 correct' is an impossible outcome |
Answer: (a) Zero.
Why each option is right or wrong
- (a) CORRECT. It is logically impossible to have exactly 5 out of 6 letters correctly placed — if 5 are correct, the 6th must be too.
- (b) Incorrect. 1/6 doesn't account for the logical impossibility of this exact scenario.
- (c) Incorrect. 1/2 also doesn't match the correct zero-probability conclusion.
- (d) Incorrect. 5/6 also doesn't match; it misreads the problem's logical constraint.
Key point
The probability of EXACTLY 5 out of 6 letters being correctly placed is ZERO — if 5 are correct, the 6th letter automatically must be correct too, by elimination, making 'exactly 5' logically impossible.
Remember
With 6 letters and 6 envelopes, the probability of exactly 5 being correctly placed is zero — it's logically impossible.
Common mistakes
- Not recognising the logical constraint: with only one letter and one envelope left, a mismatch is impossible.
- Treating this as a standard probability calculation without noticing the structural impossibility.
- Confusing 'exactly 5 correct' with 'at least 5 correct', which would behave differently.
Exam tip
'Exactly n-1 correct out of n' in a derangement-style problem is ALWAYS impossible — if n-1 are correct, the last one is forced to be correct too. A classic logical-probability insight.
Sources
- NCERT Class XI Mathematics — Probability
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Probability)