Three letters, to each of which corresponds an envelope, are placed in the envelopes at random. The probability that all the letters are not placed in the right envelopes, is

Answer

To solve the problem of finding the probability that none of the three letters are placed in the correct envelopes, we can follow these steps:

### Step 1: Understand the Total Arrangements
The total number of ways to arrange 3 letters in 3 envelopes is given by the factorial of the number of letters/envelopes.

\[ \text{Total arrangements} = 3! = 6 \]

### Step 2: Calculate the Derangements
A derangement is a permutation of elements such that none of the elements appear in their original position. We can calculate the number of derangements (denoted as \( !n \)) for \( n = 3 \) using the formula:

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

For \( n = 3 \):

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

Calculating each term:

- \( 3! = 6 \)
- \( 1 - \frac{1}{1} + \frac{1}{2} - \frac{1}{6} = 1 - 1 + 0.5 - 0.1667 = 0.3333 \)

Now, substituting back:

\[ !3 = 6 \times 0.3333 = 2 \]

### Step 3: Calculate the Probability
The probability that none of the letters are placed in the correct envelopes is given by the ratio of the number of derangements to the total arrangements.

\[ \text{Probability} = \frac{\text{Number of derangements}}{\text{Total arrangements}} = \frac{!3}{3!} = \frac{2}{6} = \frac{1}{3} \]

### Final Answer
Thus, the probability that all the letters are not placed in the right envelopes is:

\[ \frac{1}{3} \]

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