Counting the Numbers between 10 and 1000 using 2,3,4,0,8,9

There are a few steps we can take to solve this problem:

Step 1: Count the number of digits in each number
- The smallest number we can form is 20, which has two digits
- The largest number we can form is 999, which has three digits

Step 2: Count the number of possibilities for each digit
- For the first digit, we have five options (2,3,4,8,9), since we cannot use 0 as the first digit
- For the second and third digits, we have six options (0,2,3,4,8,9)

Step 3: Count the total number of possibilities
- To count the total number of possibilities, we can use the multiplication principle
- We have 5 options for the first digit, and 6 options for each of the second and third digits
- Therefore, the total number of possibilities is 5 x 6 x 6 = 180

Step 4: Eliminate numbers outside the range of 10 to 1000
- We need to eliminate numbers that are less than 10 or greater than 1000
- Numbers less than 10 have only one digit, which we cannot form using the given digits
- Numbers greater than 1000 have four or more digits, which we cannot form using the given digits
- To eliminate these numbers, we need to count the number of possibilities for each case
- For numbers less than 10, there are 0 possibilities
- For numbers greater than 1000, there are 0 possibilities
- Therefore, the total number of numbers between 10 and 1000 that can be formed using the given digits is 180 - 0 - 0 = 180

Therefore, the number of numbers lying between 10 and 1000 that can be formed with the digits 2,3,4,0,8,9 is 180.