1. **State the problem:** We want to find how many different five-letter codes can be made using the letters $a, b, c, d, e$. Then, we want to find how many of these codes start with the letters $ab$. 2. **Formula and rules:** Since repetition is allowed and order matters, the number of codes of length $n$ from $k$ letters is $k^n$. 3. **Calculate total codes:** Here, $k=5$ letters and $n=5$ positions. $$\text{Total codes} = 5^5 = 3125$$ 4. **Calculate codes starting with $ab$:** The first two letters are fixed as $a$ and $b$, so only the last three positions vary. Number of choices for each of the last three positions is 5. $$\text{Codes starting with } ab = 5^3 = 125$$ 5. **Final answers:** - Total five-letter codes: $3125$ - Codes starting with $ab$: $125$