The number of ways in which four different letters can be put in their four addressed envelopes such that atleast two of them are in the wrong envelopes are

Options

A. 23
B. 25
C. 6
D. 7

Answer

To solve the problem of finding the number of ways in which four different letters can be put in their four addressed envelopes such that at least two of them are in the wrong envelopes, we can break it down into cases based on the number of letters that are placed incorrectly.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to count the arrangements of four letters in four envelopes such that at least two letters are in the wrong envelopes.

2. **Total Arrangements**: The total arrangements of four letters in four envelopes is given by \(4!\) (factorial of 4), which equals 24.

3. **Using the Principle of Inclusion-Exclusion**: To find the arrangements where at least two letters are in the wrong envelopes, we can calculate the total arrangements and then subtract the cases where fewer than two letters are in the wrong envelopes.

4. **Case 1: No letters in wrong envelopes (All correct)**:
- There is only 1 way to arrange all letters correctly.
- Count = 1.

5. **Case 2: Exactly one letter in the wrong envelope**:
- If one letter is in the wrong envelope, the remaining three must also be in their correct envelopes.
- This is not possible because if one letter is wrong, it forces the others to be wrong as well.
- Count = 0.

6. **Calculating the Total with Inclusion-Exclusion**:
- Total arrangements = 24.
- Arrangements with fewer than two letters wrong = 1 (all correct) + 0 (one wrong) = 1.
- Arrangements with at least two letters wrong = Total arrangements - Arrangements with fewer than two wrong = \(24 - 1 = 23\).

7. **Final Count**: Therefore, the number of ways in which at least two letters are in the wrong envelopes is **23**.

### Final Answer:
The number of ways in which four different letters can be put in their four addressed envelopes such that at least two of them are in the wrong envelopes is **23**.