There are four letters and four envelopes and exactly one letter is to be put in exactly one envelope with the correct address. If the letters are randomly inserted into the envelopes, then consider the following statements : 1. It is possible that exactly one letter goes into an incorrect envelope. 2. There are only six ways in which only two letters can go into the correct envelopes. Which of the statements given above is/are correct ?
- (a) 1 only
- (b) 2 only ✓ UPSC's answer
- (c) Both 1 and 2
- (d) Neither 1 nor 2
Why the answer is (b)
• Statement 1 is incorrect because if exactly one letter is in the wrong envelope, the remaining three letters must be in the correct envelopes, which is logically impossible as the single misplaced letter would have no valid envelope to occupy.
• For Statement 2, we first choose which two letters are correctly placed from the four available, which can be done in $\binom{4}{2} = 6$ ways.
• Once two letters are fixed in their correct envelopes, the remaining two letters must be placed in the remaining two envelopes.
• To ensure that only these two are correct, the remaining two letters must be swapped (deranged), which has exactly 1 way to occur.
• Therefore, the total number of ways is $6 \times 1 = 6$, making Statement 2 correct.
• Since only Statement 2 is correct, option (b) is the right answer.
Why the other options are wrong
- (a) 1 only
- Statement 1 is false because it is impossible for exactly one letter to be in an incorrect envelope while the others are correct.
- (c) Both 1 and 2
- Statement 1 is false, so the option claiming both statements are correct is invalid.
- (d) Neither 1 nor 2
- Statement 2 is true, so the option claiming neither statement is correct is invalid.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.