There are 4 letters and 4 directed envelopes. These 4 letters are randomly inserted into the 4 envelops. What is the probability that the letters are inserted into the corresponding envelopes ?

Options

A. `11/12`
B. `23/24`
C. `1/24`
D. None of these

Explanation

To solve the problem of finding the probability that 4 letters are inserted into their corresponding envelopes, we can follow these steps:

### Step 1: Understand the Problem
We have 4 letters and 4 envelopes, and we want to find the probability that each letter ends up in the correct envelope.

### Step 2: Identify the Total Number of Arrangements
When we insert 4 letters into 4 envelopes, each letter can go into any of the envelopes. Therefore, the total number of ways to arrange the letters in the envelopes is given by the factorial of the number of letters (or envelopes).

\[ \text{Total arrangements} = 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 3: Identify the Favorable Outcomes
There is only one favorable outcome where each letter goes into its corresponding envelope. For example:
- Letter 1 goes into Envelope 1
- Letter 2 goes into Envelope 2
- Letter 3 goes into Envelope 3
- Letter 4 goes into Envelope 4

Thus, the number of favorable outcomes is 1.

### Step 4: Calculate the Probability
The probability \( P \) that all letters are inserted into their corresponding envelopes is given by the ratio of the number of favorable outcomes to the total number of arrangements.

\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total arrangements}} = \frac{1}{24} \]

### Conclusion
The probability that the letters are inserted into the corresponding envelopes is \( \frac{1}{24} \).

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