You are to create a password using 7 letters from the alphabet (repetition allowed). What is the probability that no letter is repeated if the letters were randomly chosen to be in the password? Round your answer to four decimal places.

To solve the problem, let's first break it down into steps:

1. **Total number of possible passwords**: 
   - Since repetition of letters is allowed, for each of the 7 positions in the password, we can choose any of the 26 letters in the alphabet.
   - Therefore, the total number of possible passwords (with repetition allowed) is:
     \[
     26^7
     \]

2. **Number of possible passwords with no repetition**: 
   - If no letter is repeated, the first position can have 26 options, the second position can have 25 options, the third 24 options, and so on, down to the seventh position having 20 options.
   - Therefore, the number of possible passwords with no repeated letters is:
     \[
     26 \times 25 \times 24 \times 23 \times 22 \times 21 \times 20
     \]

3. **Probability of no repetition**:
   - The probability is the ratio of the number of passwords with no repetition to the total number of passwords. That is:
     \[
     \text{Probability} = \frac{26 \times 25 \times 24 \times 23 \times 22 \times 21 \times 20}{26^7}
     \]

Now, let's calculate this value.

The probability that no letter is repeated in a randomly chosen 7-letter password is approximately \( 0.4128 \).

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. What is the probability of having exactly one repeated letter in the password?
2. How does the probability change if the password length is increased to 10 letters?
3. What is the expected number of repeated letters in a randomly chosen password?
4. How would the probability change if the alphabet was reduced to only 10 letters?
5. How can permutations be applied to generate unique passwords with constraints?

**Tip:** When calculating probabilities, always consider both the total possible outcomes and the specific favorable outcomes.

Explore how to calculate the probability that no letter is repeated in a 7-letter password randomly chosen from the 26-letter alphabet. This problem involves core concepts of probability, permutations, and combinatorics, making it suitable for students in Grades 10-12.

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