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The problem asks how many 4-letter codes can be formed using only the vowels \(a\), \(e\), \(i\), \(o\), or \(u\), with repetition allowed.

### Step-by-Step Explanation:

- There are 5 vowels to choose from: \(a\), \(e\), \(i\), \(o\), \(u\).
- Since repetition is allowed, each letter in the code can independently be one of these 5 vowels.
- The code consists of 4 letters.

Thus, for each of the 4 positions, there are 5 possible choices (one for each vowel).

The total number of possible codes is calculated as:

\[
\text{Number of codes} = 5 \times 5 \times 5 \times 5 = 5^4
\]

\[
5^4 = 625
\]

### Conclusion:
The number of possible 4-letter codes is \( \boxed{625} \).

Let me know if you would like more details or have any questions! Here are five related questions you might want to explore:

1. What if the codes had 5 letters instead of 4? How would that change the result?
2. How many 4-letter codes can be made if repetition of vowels is not allowed?
3. What if the letters could also include consonants? How many codes would be possible then?
4. How many 4-letter codes are possible if the order of the letters doesn't matter?
5. How does the number of codes change if only 3 vowels are allowed instead of 5?

**Tip:** When repetition is allowed in combinatorics, multiply the number of options available for each position in the sequence!

How many 4-letter codes are possible to make using only vowels, a, e, i, o, or u? (Repeated letters are okay here.)

Explore how to calculate the total number of 4-letter codes made from the vowels a, e, i, o, and u with repetition allowed. This combinatorics problem introduces the multiplication principle and uses powers of 5 to find the solution.

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