1. **Problem statement:** How many different identification codes can be made if the code consists of 2 numbers followed by 5 letters, the code cannot begin with 0, cannot contain the letter O, and repetitions of numbers and letters are not allowed. 2. **Understanding the code format:** The code is of the form NNLLLLL, where N represents a number and L represents a letter. 3. **Numbers:** The first number cannot be 0, so it can be any digit from 1 to 9, giving 9 options. 4. The second number cannot be the same as the first (no repetition), so it has 8 remaining options. 5. **Letters:** The letters cannot include O, so from the 26 letters of the alphabet, we exclude O, leaving 25 letters. 6. Since repetitions are not allowed, the first letter has 25 options, the second 24, the third 23, the fourth 22, and the fifth 21. 7. **Calculate total number of codes:** $$\text{Total} = 9 \times 8 \times 25 \times 24 \times 23 \times 22 \times 21$$ 8. Calculate step-by-step: $$9 \times 8 = 72$$ $$25 \times 24 = 600$$ $$600 \times 23 = 13,800$$ $$13,800 \times 22 = 303,600$$ $$303,600 \times 21 = 6,375,600$$ 9. Multiply numbers and letters parts: $$72 \times 6,375,600 = 459,043,200$$ **Final answer:** There are 459,043,200 different identification codes possible under the given conditions.