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?

Question

CSATEasyPrelims 2023Maths

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:

It is possible that exactly one letter goes into an incorrect envelope.
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?

Explanation

Statement 1: If one letter is placed in an incorrect envelop, then the letter that belonged to that envelop must have also been placed in an incorrect envelop. So, it's not possible that only one letter gets placed in an incorrect envelop. Either no letter will get misplaced, or at least two letters will get misplaced.

So, Statement 1 is incorrect.

Statement 2: We need to place two letters in correct envelops and two in incorrect envelops. We can choose the two letters to be placed in correct envelops in $4C2$ ways, i.e. $6$ ways. The remaining two letters will automatically get chosen to get placed in each other's envelops (i.e. incorrect envelops).