What is the probability that if 15 letters are typed, no letters are repeated? Please provide your answer as a fraction, or with at least 4 decimals.

The probability that no letters are repeated is

To find the probability that no letters are repeated when 15 letters are typed, we can calculate it using the principles of probability and combinatorics.

### Step-by-Step Solution:

1. **Total possible combinations:**
   There are 26 letters in the alphabet. If 15 letters are typed, and there are no restrictions, each of the 15 letters could be any of the 26 letters. The total number of ways to type 15 letters (with repetitions allowed) is:
   \[
   26^{15}
   \]

2. **Combinations with no repetition:**
   If no letters are repeated, the first letter can be any of the 26 letters, the second letter can be any of the remaining 25 letters, the third letter can be any of the remaining 24 letters, and so on. The number of ways to type 15 letters without repetition is:
   \[
   26 \times 25 \times 24 \times \dots \times (26 - 15 + 1) = \frac{26!}{(26 - 15)!} = \frac{26!}{11!}
   \]

3. **Probability:**
   The probability that no letters are repeated is the ratio of the number of ways to type 15 letters without repetition to the total number of ways to type 15 letters (with repetition). Thus, the probability is:
   \[
   P(\text{no repetition}) = \frac{26 \times 25 \times 24 \times \dots \times 12}{26^{15}}
   \]

Let’s calculate this value.The probability that no letters are repeated when 15 letters are typed is approximately \( 0.0060 \), or about \( \frac{6}{1000} \).

Would you like more details on the steps, or do you have any other questions?

Here are 5 related questions:
1. What is the probability that no letters are repeated when typing 10 letters?
2. How does the probability change if 20 letters are typed?
3. Can we calculate the probability that exactly one letter repeats in 15 typed letters?
4. How does the inclusion of uppercase and lowercase letters change the probability?
5. What is the expected number of unique letters in 15 randomly typed letters?

**Tip:** In combinatorics, calculating probabilities often involves dividing specific favorable outcomes by the total possible outcomes.

What is the probability that if 15 letters are typed, no letters are repeated?

Explore the probability of typing 15 letters without any repetitions. This solution breaks down the problem using combinatorics and probability theory, suitable for high school students in Grades 10-12.

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