There are four letters and four envelopes, the letters are placed into the envelopes at random, find the probability that all letters are placed in the wrong envelopes.

Answer

To solve the problem of finding the probability that all four letters are placed in the wrong envelopes, we can follow these steps:

### Step 1: Understand the Total Arrangements
First, we need to calculate the total number of ways to arrange 4 letters into 4 envelopes. This can be done using the factorial of the number of letters/envelopes.

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

### Step 2: Calculate Derangements
Next, we need to find the number of arrangements where none of the letters are placed in their correct envelopes. This is known as a derangement.

The formula for the number of derangements \( !n \) of \( n \) items is given by:

\[ !n = n! \sum_{i=0}^{n} \frac{(-1)^i}{i!} \]

For \( n = 4 \):

\[ !4 = 4! \left( \frac{(-1)^0}{0!} + \frac{(-1)^1}{1!} + \frac{(-1)^2}{2!} + \frac{(-1)^3}{3!} + \frac{(-1)^4}{4!} \right) \]

Calculating each term:

- \( \frac{(-1)^0}{0!} = 1 \)
- \( \frac{(-1)^1}{1!} = -1 \)
- \( \frac{(-1)^2}{2!} = \frac{1}{2} \)
- \( \frac{(-1)^3}{3!} = -\frac{1}{6} \)
- \( \frac{(-1)^4}{4!} = \frac{1}{24} \)

Now summing these:

\[ 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} \]

Finding a common denominator (24):

\[ = \frac{24}{24} - \frac{24}{24} + \frac{12}{24} - \frac{4}{24} + \frac{1}{24} = \frac{9}{24} = \frac{3}{8} \]

Now, multiply by \( 4! \):

\[ !4 = 24 \times \frac{9}{24} = 9 \]

### Step 3: Calculate the Probability
Now that we have the number of derangements, we can find the probability that all letters are placed in the wrong envelopes.

\[ \text{Probability} = \frac{\text{Number of derangements}}{\text{Total arrangements}} = \frac{!4}{4!} = \frac{9}{24} = \frac{3}{8} \]

### Final Answer
Thus, the probability that all letters are placed in the wrong envelopes is:

\[ \frac{3}{8} \]

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