There are four letters and four envelopes and exactly one letter is to be put in exactly one envelope with…
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?
1 only
2 only
Both 1 and 2
Neither 1 nor 2
Solution
Statement 1 is false: it is impossible for exactly one letter to be in the wrong envelope, because if three are correct the fourth must also be correct. Statement 2: the number of ways exactly two letters are correct is $\binom{4}{2} \times D_2 = 6 \times 1 = 6$, where $D_2 = 1$ is the number of derangements of the remaining 2. So statement 2 is correct. Only statement 2 is correct.