# Q. 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?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Correct Answer: (b) 2 only

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

**The first statement is incorrect.**

## If we have four letters and four envelopes, and we place one letter correctly, then we have three letters and three envelopes left.

If we now place another letter correctly, we would have two letters and two envelopes left, which means either both of these would be correct or both would be incorrect. If we place one of these two letters incorrectly, the last letter would also have to go into the incorrect envelope because there’s only one choice left. Therefore, it’s not possible to have exactly one letter in an incorrect envelope. The possible scenarios are 0, 2, or 4 letters in the correct envelopes.

**The second statement is correct.**

### There are indeed six ways in which only two letters can go into the correct envelopes.

This can be calculated using the combination formula 4C2 (which stands for “”4 choose 2″”), which gives us the number of ways to choose 2 items from a set of 4. The formula is 4! / (2!(4-2)!) = 6.

So, the correct answer is (b) 2 only.