1. **Problem statement:** At Santa’s workshop, each Elf's ID consists of 2 numbers followed by 5 letters. The code cannot begin with 0, cannot contain the letter O, and no repetitions of numbers or letters are allowed. How many different identification codes can be made? 2. **Understanding the problem:** - The ID format is: NNLLLLL (2 numbers, 5 letters). - The first number cannot be 0. - The letter O is not allowed. - No repetition of numbers or letters. 3. **Step 1: Counting the numbers** - Numbers are digits 0-9, so 10 digits total. - First digit cannot be 0, so choices for first digit: 9 (digits 1-9). - Second digit cannot be the first digit (no repetition), so choices: 9 (digits 0-9 except the first digit). 4. **Step 2: Counting the letters** - Letters are A-Z, 26 total. - Letter O is not allowed, so letters available: 25. - No repetition in letters, so for 5 letters: - 1st letter: 25 choices - 2nd letter: 24 choices - 3rd letter: 23 choices - 4th letter: 22 choices - 5th letter: 21 choices 5. **Step 3: Calculate total number of codes** - Total = (choices for numbers) * (choices for letters) - Numbers: $9 \times 9 = 81$ - Letters: $25 \times 24 \times 23 \times 22 \times 21$ 6. **Step 4: Multiply all together** $$ \text{Total codes} = 81 \times 25 \times 24 \times 23 \times 22 \times 21 $$ 7. **Step 5: Calculate the product** - Calculate letters part first: $25 \times 24 = 600$ $600 \times 23 = 13,800$ $13,800 \times 22 = 303,600$ $303,600 \times 21 = 6,375,600$ - Now multiply by numbers: $6,375,600 \times 81 = 516,355,600$ **Final answer:** There are $516,355,600$ different identification codes possible under the given conditions.