In how many ways we can put 5 writings into 5 corresponding envelopes so that no writing go to the corresponding envelope?

Answer

To solve the problem of how many ways we can put 5 writings into 5 corresponding envelopes such that no writing goes into its corresponding envelope, we will use the concept of derangements. A derangement is a permutation of elements such that none of the elements appear in their original position.

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 5 writings (let's denote them as W1, W2, W3, W4, W5) and 5 corresponding envelopes (E1, E2, E3, E4, E5). We need to find the number of ways to arrange these writings in the envelopes such that no writing is placed in its corresponding envelope.

2. **Using the Derangement Formula**: The number of derangements (denoted as !n) of n items can be calculated using the formula:
\[ !n = n! \left( \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + \frac{(-1)^n}{n!} \right) \]
For our case, n = 5.

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

4. **Calculating the Series**: Now we compute the series:
\[ \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} \]
- \( \frac{1}{0!} = 1 \)
- \( \frac{1}{1!} = 1 \)
- \( \frac{1}{2!} = \frac{1}{2} \)
- \( \frac{1}{3!} = \frac{1}{6} \)
- \( \frac{1}{4!} = \frac{1}{24} \)
- \( \frac{1}{5!} = \frac{1}{120} \)

Plugging these values into the series:
\[ 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} - \frac{1}{120} \]

5. **Finding a Common Denominator**: The least common multiple of the denominators (1, 1, 2, 6, 24, 120) is 120. Now we convert each term:
- \( 1 = \frac{120}{120} \)
- \( -1 = \frac{-120}{120} \)
- \( \frac{1}{2} = \frac{60}{120} \)
- \( -\frac{1}{6} = \frac{-20}{120} \)
- \( \frac{1}{24} = \frac{5}{120} \)
- \( -\frac{1}{120} = \frac{-1}{120} \)

Now summing these fractions:
\[ \frac{120 - 120 + 60 - 20 + 5 - 1}{120} = \frac{44}{120} \]

6. **Calculating the Derangement**: Now we multiply this result by 5!:
\[ !5 = 120 \times \frac{44}{120} = 44 \]

### Final Answer:
The total number of ways to arrange the writings in the envelopes such that no writing goes into its corresponding envelope is **44**.