(i) To find the number of three-digit numbers with all different digits, consider the hundreds, tens, and units places:
- The hundreds digit can be any digit from 1 to 9 (9 options).
- The tens digit can be any digit except the hundreds digit (9 options).
- The units digit can be any digit except the hundreds and tens digits (8 options).
Thus, the total number of such numbers is:
\(9 \times 9 \times 8 = 648\)
(ii) To find how many of these numbers are odd and greater than 700:
- Consider numbers starting with 7, 8, or 9.
- For numbers starting with 7, the units digit must be odd (1, 3, 5, 7, 9), giving 4 options (since 7 is already used).
- The tens digit can be any of the remaining 8 digits.
- Total for numbers starting with 7: \(1 \times 8 \times 4 = 32\)
- For numbers starting with 8, the units digit must be odd (1, 3, 5, 7, 9), giving 5 options.
- The tens digit can be any of the remaining 8 digits.
- Total for numbers starting with 8: \(1 \times 8 \times 5 = 40\)
- For numbers starting with 9, the units digit must be odd (1, 3, 5, 7, 9), giving 4 options (since 9 is already used).
- The tens digit can be any of the remaining 8 digits.
- Total for numbers starting with 9: \(1 \times 8 \times 4 = 32\)
Adding these gives the total number of odd numbers greater than 700:
\(32 + 40 + 32 = 104\)
