To log on to a certain computer account, the user must type in a 5-letter password. In such a password, no letter may be repeated, and only the lower case of a letter may be used. How many such 5-letter passwords are possible? (There are 26 letters in the alphabet.)

To determine how many 5-letter passwords can be formed under the given conditions, we need to consider the following points:

1. **Each letter must be unique** (no repetition).
2. **Only lowercase letters** (26 total letters: a to z) are allowed.

### Step-by-Step Calculation:

1. **First letter**: Since the password consists of lowercase letters, we have 26 possible choices for the first letter.
   
2. **Second letter**: For the second letter, it must be different from the first. Therefore, we have 25 remaining choices.
   
3. **Third letter**: The third letter must differ from the first two, leaving us with 24 choices.
   
4. **Fourth letter**: The fourth letter must be different from the first three, so we have 23 choices.
   
5. **Fifth letter**: The fifth and final letter must be different from the first four, leaving us with 22 choices.

### Total Number of Possible Passwords:
To find the total number of possible 5-letter passwords, we multiply the number of choices for each letter:

\[
26 \times 25 \times 24 \times 23 \times 22
\]

Let's calculate this product:

\[
26 \times 25 = 650
\]
\[
650 \times 24 = 15600
\]
\[
15600 \times 23 = 358800
\]
\[
358800 \times 22 = 7893600
\]

Thus, the total number of possible 5-letter passwords is:

\[
\boxed{7,893,600}
\]

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:
1. How many 4-letter passwords can be created under similar conditions?
2. What if the password could include both uppercase and lowercase letters?
3. How would the result change if repetition of letters was allowed?
4. How many 6-letter passwords can be created using the same conditions?
5. What if the password included numbers (0-9) along with letters?

**Tip:** When counting permutations of distinct objects, the number of choices decreases with each selection!

Discover how to calculate the total number of possible 5-letter passwords using combinatorics. This problem explores permutation principles with no repetition and lowercase letters, resulting in 7,893,600 possible passwords. Ideal for students in Grades 9-12.

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