(a) The number of ways to choose 2 letters from 5 is given by permutations: \(^5P_2 = 5 \times 4\). The number of ways to choose 4 digits from 7 is \(^7P_4 = 7 \times 6 \times 5 \times 4\). Therefore, the total number of codes is \(5 \times 4 \times 7 \times 6 \times 5 \times 4 = 16,800\).

(b) We use the principle of inclusion-exclusion. Calculate the number of codes with A, with 5, and with both:

1. Codes with A and no 5: \(8 \times ^6P_4 = 2,880\).
2. Codes with 5 and no A: \(4 \times ^6P_3 = 5,760\).
3. Codes with both A and 5: \(8 \times ^5P_3 = 3,840\).

Total codes with A or 5 or both: \(2,880 + 5,760 + 3,840 = 12,480\).

(c) The number of codes that start with DE and have a number between 4500 and 5000 is calculated as follows:

There is only one way to choose DE. The number must be between 4500 and 5000, so the first digit is 4, and the remaining digits are chosen from \{5, 6, 7\}:

Number of ways to choose the remaining 3 digits: \(^3P_3 = 6\).

Total successful codes: \(1 \times 6 = 6\).

Probability: \(\frac{6}{16,800} = \frac{1}{280}\).