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 wrong: if three letters are in their correct envelopes, the only envelope left is the fourth letter's own, so exactly one letter can never be misplaced.
• Statement 2 is correct: the two letters that go right can be chosen in 4C2 = 6 ways, and the other two must then be swapped, which can happen in only one way.
• So there are exactly 6 ways in which only two letters reach the correct envelopes.
• Hence 2 only, option (b).
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.