The number of ways in which 5 letters can be placed in 5 marked envelopes, so that no letter is in the right encelope, is

Options

A. 45
B. 44
C. 43
D. 46

Explanation

To solve the problem of finding the number of ways in which 5 letters can be placed in 5 marked envelopes such that no letter is in the correct envelope, we will use the concept of dearrangements. A dearrangement is a permutation of the elements of a set, such that none of the elements appear in their original position.

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the number of ways to arrange 5 letters in 5 envelopes such that no letter goes into its corresponding envelope. This is a classic example of a dearrangement.

2. **Use the Formula for Dearrangement**: The number of dearrangements \( !n \) of \( n \) items can be calculated using the formula:
\[ !n = n! \sum_{i=0}^{n} \frac{(-1)^i}{i!} \]
For our case, \( n = 5 \).

3. **Calculate \( 5! \)**: First, we need to calculate \( 5! \):
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

4. **Calculate the Summation**:
We will compute the summation \( \sum_{i=0}^{5} \frac{(-1)^i}{i!} \):
- For \( i = 0 \): \( \frac{(-1)^0}{0!} = 1 \)
- For \( i = 1 \): \( \frac{(-1)^1}{1!} = -1 \)
- For \( i = 2 \): \( \frac{(-1)^2}{2!} = \frac{1}{2} \)
- For \( i = 3 \): \( \frac{(-1)^3}{3!} = -\frac{1}{6} \)
- For \( i = 4 \): \( \frac{(-1)^4}{4!} = \frac{1}{24} \)
- For \( i = 5 \): \( \frac{(-1)^5}{5!} = -\frac{1}{120} \)

Now, we sum these values:
\[ 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} - \frac{1}{120} \]

To simplify this, we can find a common denominator, which is 120:
\[ = \frac{120}{120} - \frac{120}{120} + \frac{60}{120} - \frac{20}{120} + \frac{5}{120} - \frac{1}{120} \]
\[ = 0 + \frac{60 - 20 + 5 - 1}{120} = \frac{44}{120} \]

5. **Final Calculation**:
Now, substituting back into the dearrangement formula:
\[ !5 = 5! \cdot \left(\frac{44}{120}\right) = 120 \cdot \frac{44}{120} = 44 \]

Thus, the number of ways to arrange 5 letters in 5 envelopes such that no letter is in the correct envelope is **44**.