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?
Answer & explanation
Answer: (b) 2 only
If three letters are in their correct envelopes, the only envelope left belongs to the fourth letter, so exactly one wrong letter is impossible. For exactly two correct, choose the two correct letters in 6 ways and swap the other two in 1 way: 6 ways.
- Statement 1: if 3 letters are correct, the 4th letter is left with only its own envelope, so it is correct too. Exactly one wrong letter cannot happen.
- Statement 2: choose the 2 letters that go correctly: C(4, 2) = 6 ways.
- The other 2 letters must both be wrong, which is possible in only 1 way (they swap envelopes).
- Ways = 6 × 1 = 6, so statement 2 is correct.
- ✗ 1. With three letters correctly placed, the last envelope is the fourth letter's own; one misplaced letter alone is impossible.
- ✓ 2. C(4, 2) = 6 choices of the two correct letters, and the remaining two can be wrong in just one way (a swap): 6 ways.
Remember · Exactly one item misplaced is never possible. 'Exactly k correct' = C(n, k) × ways to put all the rest wrong.
Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·