Math Problem Statement
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.)
Solution
To determine how many 5-letter passwords can be formed under the given conditions, we need to consider the following points:
- Each letter must be unique (no repetition).
- Only lowercase letters (26 total letters: a to z) are allowed.
Step-by-Step Calculation:
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First letter: Since the password consists of lowercase letters, we have 26 possible choices for the first letter.
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Second letter: For the second letter, it must be different from the first. Therefore, we have 25 remaining choices.
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Third letter: The third letter must differ from the first two, leaving us with 24 choices.
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Fourth letter: The fourth letter must be different from the first three, so we have 23 choices.
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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:
Let's calculate this product:
Thus, the total number of possible 5-letter passwords is:
Would you like more details or have any questions?
Here are 5 related questions to expand your understanding:
- How many 4-letter passwords can be created under similar conditions?
- What if the password could include both uppercase and lowercase letters?
- How would the result change if repetition of letters was allowed?
- How many 6-letter passwords can be created using the same conditions?
- 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!
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Permutations
Formulas
P(n, k) = n! / (n - k)!
Theorems
Permutation Theorem
Suitable Grade Level
Grades 9-12
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