A person writes letters to six friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that all of them are in a wrong envelope

Explanation

To solve the problem of how many ways a person can place six letters into six envelopes such that none of the letters is in the correct envelope (a derangement), we can use the formula for derangements.

### Step-by-Step Solution:

1. **Understanding Derangement**: A derangement is a permutation of the elements such that none of the elements appear in their original position. We denote the number of derangements of n items as \( !n \).

2. **Derangement Formula**: The formula for the number of derangements \( !n \) is given by:
\[
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