Given: Digits 2,3,4,0,8,9

To find: Number of numbers lying between 10 and 1000 that can be formed using these digits.

Solution:

1. Single digit numbers:

The given digits are 2,3,4,0,8,9. Out of these, 0 cannot be used as the first digit of a number. Hence, the single digit numbers that can be formed are:

2, 3, 4, 8, 9

Total count = 5

2. Two-digit numbers:

We need to find the number of two-digit numbers that can be formed using the given digits. We can use the permutation formula for this.

Number of two-digit numbers = P(5,2) = 5 x 4 = 20

However, we need to exclude the two-digit numbers that are less than 10.

Number of two-digit numbers less than 10 = 1 (i.e. 0)

Hence, the number of two-digit numbers that can be formed using the given digits = 20 - 1 = 19

3. Three-digit numbers:

We need to find the number of three-digit numbers that can be formed using the given digits. Again, we can use the permutation formula.

Number of three-digit numbers = P(5,3) = 5 x 4 x 3 = 60

However, we need to exclude the three-digit numbers that are less than 100.

Number of three-digit numbers less than 100 = P(4,2) = 4 x 3 = 12

Hence, the number of three-digit numbers that can be formed using the given digits = 60 - 12 = 48

4. Total number of numbers:

The total number of numbers that can be formed using the given digits = (Number of single digit numbers) + (Number of two-digit numbers) + (Number of three-digit numbers)

= 5 + 19 + 48

= 72

However, we need to exclude the numbers less than 10.

Number of numbers less than 10 = 1 (i.e. 0)

Hence, the final answer = 72 - 1 = 71.

Therefore, the given answer is incorrect.

CORRECT ANSWER: None of these.