There are 10 envelopes and 10 letters to go inside them. Each letter is meant for a specified envelope only. What is the probability that exactly 9 of them are in the right envelopes ?

Options

A. `1//10!`
B. 1
C. 0
D. None of these

Answer

To solve the problem of finding the probability that exactly 9 out of 10 letters are placed in the correct envelopes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 10 letters and 10 envelopes. Each letter corresponds to one specific envelope. We need to determine the probability that exactly 9 letters are placed in their correct envelopes.

2. **Analyzing the Situation**: If 9 letters are in the correct envelopes, that means the 10th letter must also be in its correct envelope. This is because there is only one envelope left for the last letter, which must be the correct one. Therefore, it is impossible to have exactly 9 letters in the correct envelopes without the 10th letter also being correct.

3. **Conclusion**: Since it is impossible to have exactly 9 letters in the correct envelopes while the 10th letter is not, the number of favorable outcomes (where exactly 9 letters are correct) is 0.

4. **Total Outcomes**: The total number of ways to arrange 10 letters in 10 envelopes is given by the factorial of 10, which is \(10!\).

5. **Calculating Probability**: The probability \(P\) of an event is given by the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
Here, the number of favorable outcomes is 0, and the total number of outcomes is \(10!\). Thus,
\[ P = \frac{0}{10!} = 0 \]

### Final Answer:
The probability that exactly 9 letters are in the right envelopes is \(0\).