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?

The correct answer is (B) 2 only.

Explanation:

Let's evaluate the two statements based on derangement principles (subfactorials) for n = 4 letters and envelopes. Statement 1: 'It is possible that exactly one letter goes into an incorrect envelope.' In combinatorial derangements, a letter placed in an envelope either goes to the correct one or an incorrect one. If n-1 letters are in their correct envelopes, the remaining 1 letter MUST also be in its correct envelope (since no single letter can be misplaced alone without leaving another misplaced). Thus, it is mathematically impossible for exactly one letter to be incorrect (or correct); misplacements always occur in groups of at least two. Hence, Statement 1 is incorrect. Statement 2: 'There are only six ways in which only two letters can go into the correct envelopes.' The number of ways to choose which 2 letters go into the correct envelopes out of 4 is C(4, 2) = 6. For the remaining 2 letters, they must both go into incorrect envelopes (i.e., a derangement of 2 objects, which has D(2) = 1 way). Thus, total ways = 6 × 1 = 6 ways. Statement 2 is correct. Therefore, only statement 2 is correct.