Let's break down the problem step by step.

Part (a)(i)

We are tasked with finding how many numbers between 100 and 999 have all three digits different. These are three-digit numbers, so the first digit (hundreds place) cannot be 0, and each digit must be distinct.

• For the hundreds digit, we have 9 options: 1 to 9.
• For the tens digit, we have 9 options as well: 0 to 9, excluding the digit used for the hundreds place.
• For the units digit, we have 8 remaining options: excluding the digits used for the hundreds and tens places.

So, the total number of three-digit numbers with distinct digits is: $9 \times 9 \times 8 = 648$

Thus, the answer to part (a)(i) is 648.

Part (a)(ii)

Now, we are asked to find how many of the numbers in part (i) are odd numbers greater than 700.

For this, the number must be:

1. Greater than 700, so the hundreds digit must be between 7 and 9 (inclusive).
2. Odd, so the units digit must be one of the odd digits: 1, 3, 5, 7, 9.

Let’s consider each case for the hundreds digit:

• If the hundreds digit is 7:

  • The tens digit can be any digit from 0 to 9, except for 7 (so 9 options).
  • The units digit must be odd, and different from the hundreds and tens digits. We have 4 choices (1, 3, 5, 9).

  So, for hundreds digit 7, the total is:
  $1 \times 9 \times 4 = 36$

• If the hundreds digit is 8:

  • The tens digit can be any digit from 0 to 9, except for 8 (so 9 options).
  • The units digit must be odd, and different from the hundreds and tens digits. We have 5 choices (1, 3, 5, 7, 9).

  So, for hundreds digit 8, the total is:
  $1 \times 9 \times 5 = 45$

• If the hundreds digit is 9:

  • The tens digit can be any digit from 0 to 9, except for 9 (so 9 options).
  • The units digit must be odd, and different from the hundreds and tens digits. We have 5 choices (1, 3, 5, 7, 9).

  So, for hundreds digit 9, the total is:
  $1 \times 9 \times 5 = 45$

Adding these up gives: $36 + 45 + 45 = 126$

Thus, the answer to part (a)(ii) is 126.

Let me know if you'd like further explanations or assistance with similar problems!

Related Questions:

1. How would the calculation change if the digits had to repeat?
2. How many numbers between 100 and 999 have at least one repeated digit?
3. How many even numbers between 100 and 999 have distinct digits?
4. How can we calculate the number of numbers greater than 500 with all digits different?
5. How would the solution change if we required the tens digit to be a multiple of 5?

Tip: When solving such counting problems, always consider each digit place one at a time, keeping track of the restrictions (like distinct digits or number properties like odd/even) as you go!