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 determining the number of ways in which four different letters can be placed in their four addressed envelopes such that at least two of them are in the wrong envelopes, we can follow these steps:

### Step 1: Calculate the Total Arrangements
First, we need to find the total number of ways to arrange the four letters in the four envelopes. Since all letters are different, the total arrangements can be calculated using the factorial of the number of letters.

\[ \text{Total arrangements} = 4! = 24 \]

### Step 2: Calculate the Correct Arrangements
Next, we need to consider the arrangements where all letters are in their correct envelopes. There is only one way to do this, which is when each letter is placed in its corresponding envelope.

\[ \text{Correct arrangements} = 1 \]

### Step 3: Calculate the Arrangements with At Least Two Wrong
To find the number of arrangements where at least two letters are in the wrong envelopes, we can use the principle of complementary counting. We subtract the number of arrangements where all letters are correct from the total arrangements.

\[ \text{Arrangements with at least 2 wrong} = \text{Total arrangements} - \text{Correct arrangements} \]
\[ = 24 - 1 = 23 \]

### Conclusion
Thus, 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:

\[ \boxed{23} \]

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