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 calculate how many possible 5-letter passwords can be created using the given constraints (no repeated letters, and only lowercase letters), we can approach the problem step by step.

### Step-by-Step Calculation:

1. **Number of Choices for the First Letter**:
   - Since the password must be a 5-letter combination with no repetitions, for the first letter, there are **26 possible choices** (since there are 26 letters in the alphabet).

2. **Number of Choices for the Second Letter**:
   - The second letter must be different from the first, so there are now **25 remaining choices**.

3. **Number of Choices for the Third Letter**:
   - The third letter must be different from the first two letters, leaving us with **24 choices**.

4. **Number of Choices for the Fourth Letter**:
   - The fourth letter must be different from the first three letters, giving us **23 choices**.

5. **Number of Choices for the Fifth Letter**:
   - Finally, the fifth letter must be different from the first four letters, so there are **22 choices** left.

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

\[
26 \times 25 \times 24 \times 23 \times 22 = 7,893,600
\]

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

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

Would you like further details or have any questions about this process?

Here are 5 related questions for further exploration:
1. How would the calculation change if uppercase letters were also allowed?
2. How many 4-letter passwords could be created under the same conditions?
3. What is the probability of randomly guessing the correct password on the first try?
4. How many passwords can be created if repetition of letters is allowed?
5. How would the result change if only vowels were used for the password?

**Tip**: When dealing with permutations (like this case with no repetition), remember to reduce the number of choices for each subsequent selection!

Explore how to calculate the total number of possible 5-letter passwords where no letter is repeated, and only lowercase letters are used. This problem covers key combinatorics concepts, including permutations and the multiplication rule of counting. Suitable for students in Grades 8-10.

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