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 find the number of possible 5-letter passwords, where no letter is repeated and only lowercase letters are used, we can break the problem into steps based on how the letters are selected.

1. **First Letter**: There are 26 letters in the alphabet. The first letter can be any of these 26.
2. **Second Letter**: Since no letter can be repeated, the second letter can be chosen from the remaining 25 letters.
3. **Third Letter**: After choosing the first two letters, there are 24 remaining choices for the third letter.
4. **Fourth Letter**: There are 23 remaining choices for the fourth letter.
5. **Fifth Letter**: Finally, there are 22 remaining choices for the fifth letter.

The total number of possible passwords is the product of these choices:
\[
26 \times 25 \times 24 \times 23 \times 22
\]

Now, let's calculate the 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 on permutations or combinations, or have any other questions?

Here are 5 related questions to explore:
1. How would the result change if repetition of letters were allowed?
2. What if the password length were 6 letters instead of 5?
3. What if both uppercase and lowercase letters were allowed, but still no repetition?
4. How many passwords are possible if only vowels are allowed, with no repetition?
5. How does this problem relate to permutations in combinatorics?

**Tip**: When letters or digits are restricted in any counting problem, the concept of permutations (ordered selections) is often key to solving it efficiently!

Learn how to calculate the number of possible 5-letter passwords using combinatorics and permutation principles. This problem involves selecting from 26 letters with no repetition, providing a total of 7,893,600 unique password possibilities. Suitable for students in Grades 9-12.

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