1. **Problem Statement:** Alice has a safe secured by a 4-digit code with no repeated digits. We want to find the number of possible codes that satisfy this condition. 2. **Formula and Rules:** Since the code is 4 digits long and no digit repeats, this is a permutation problem where order matters and repetition is not allowed. 3. **Step-by-step Solution:** - The first digit can be any of the 10 digits (0 through 9), so there are 10 choices. - The second digit cannot be the first digit, so there are 9 choices left. - The third digit cannot be the first two digits, so there are 8 choices left. - The fourth digit cannot be any of the first three digits, so there are 7 choices left. 4. **Calculate the total number of codes:** $$10 \times 9 \times 8 \times 7 = 5040$$ 5. **Explanation:** We multiply the number of choices for each digit because each choice is independent and sequential, and digits cannot repeat. **Final answer:** There are $5040$ possible 4-digit codes with no repeated digits.