Q36. 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?

This problem involves permutations and derangements. Statement 1: 'It is possible that exactly one letter goes into an incorrect envelope.' If one letter is in an incorrect envelope, say Letter A is in Envelope B, then Envelope A cannot contain Letter A.

It must contain some other letter (B, C, or D). This means at least two letters must be misplaced. It's impossible for only one letter to be misplaced while all others are correctly placed.

Thus, Statement 1 is incorrect. Statement 2: 'There are only six ways in which only two letters can go into the correct envelopes.' We need to choose 2 letters out of 4 to be placed correctly.

This can be done in C(4, 2) ways = 6 ways. For the remaining 2 letters, they must be placed in incorrect envelopes (deranged). The number of derangements for 2 items, D(2), is 1 (they must swap positions).

So, the total number of ways is C(4, 2) * D(2) = 6 * 1 = 6 ways. Thus, Statement 2 is correct. This question tests understanding of permutations, combinations, and derangements, which are relevant for CSAT.