1. **Problem statement:** Freddie has a 6-digit code with no repeated digits and the first digit is not zero. (i) Find the number of such 6-digit numbers. (ii) Find the number of such 6-digit numbers divisible by 5. 2. **Step (i): Number of 6-digit numbers with no repeated digits and first digit not zero** - The first digit can be any digit from 1 to 9 (9 options). - The second digit can be any digit except the first digit (9 options, including zero now). - The third digit can be any digit except the first two digits (8 options). - The fourth digit: 7 options. - The fifth digit: 6 options. - The sixth digit: 5 options. So total numbers = $9 \times 9 \times 8 \times 7 \times 6 \times 5$ 3. **Step (ii): Number of such 6-digit numbers divisible by 5** - A number divisible by 5 ends with 0 or 5. - Case 1: Last digit is 0 - First digit: 9 options (1-9) - Last digit fixed as 0 - Remaining 4 digits: choose from 8 remaining digits (excluding first digit and 0) - So number of ways = $9 \times P(8,4) = 9 \times 8 \times 7 \times 6 \times 5$ - Case 2: Last digit is 5 - First digit: 8 options (1-9 except 5) - Last digit fixed as 5 - Remaining 4 digits: choose from 8 remaining digits (excluding first digit and 5) - Number of ways = $8 \times P(8,4) = 8 \times 8 \times 7 \times 6 \times 5$ - Total numbers divisible by 5 = Case 1 + Case 2 4. **Calculations:** - Step (i): $9 \times 9 \times 8 \times 7 \times 6 \times 5 = 136080$ - Step (ii): - Case 1: $9 \times 8 \times 7 \times 6 \times 5 = 15120$ - Case 2: $8 \times 8 \times 7 \times 6 \times 5 = 13440$ - Total = $15120 + 13440 = 28560$ **Final answers:** (i) $136080$ (ii) $28560$