Introduction:
To find the number of numbers that can be formed with the given digits (2, 3, 4, 0, 8, 9) lying between 10 and 1000, we need to understand the rules and conditions that apply.

Explanation:
To solve this problem, we can break it down into smaller steps:

1. Identify the range:
We need to find the numbers lying between 10 and 1000. This means our numbers should be greater than or equal to 10 and less than or equal to 1000.

2. Identify the position of digits:
The numbers we form can have 3 digits (hundreds, tens, and units place). We need to consider the position of each digit and the possible options for each position.

3. Determine the options for each position:
For the units place, we can use any of the given digits (2, 3, 4, 0, 8, 9).

For the tens place, we can use any of the given digits except 0.

For the hundreds place, we can use any of the given digits except 0 and 2.

4. Calculate the number of possibilities:
To find the total number of possibilities, we need to multiply the number of options for each position.

For the units place, we have 6 options (2, 3, 4, 0, 8, 9).

For the tens place, we have 5 options (2, 3, 4, 8, 9).

For the hundreds place, we have 4 options (3, 4, 8, 9).

Hence, the total number of possibilities is calculated as: 6 * 5 * 4 = 120.

5. Conclusion:
Therefore, the number of numbers that can be formed with the given digits (2, 3, 4, 0, 8, 9) lying between 10 and 1000 is 120.