How many numbers each lying between 9 and 1000 can be formed with the digits 0,1,2,3,7,8 ( number can be repeated ) ?

Options

A. 30
B. 120
C. 210
D. 90

Answer

To solve the problem of how many numbers can be formed between 9 and 1000 using the digits 0, 1, 2, 3, 7, and 8 (with repetition allowed), we will break it down into two parts: two-digit numbers and three-digit numbers.

### Step 1: Count the Two-Digit Numbers
1. **Identify the Range**: The two-digit numbers we can form must be between 10 and 99.
2. **First Digit Choices**: The first digit cannot be 0 (to ensure it's a two-digit number). Therefore, the possible choices for the first digit are 1, 2, 3, 7, or 8. This gives us **5 options**.
3. **Second Digit Choices**: The second digit can be any of the available digits (0, 1, 2, 3, 7, 8). This gives us **6 options**.
4. **Total Two-Digit Numbers**: The total number of two-digit numbers can be calculated by multiplying the number of choices for the first digit by the number of choices for the second digit:
\[ \text{Total Two-Digit Numbers} = 5 \times 6 = 30 \]

### Step 2: Count the Three-Digit Numbers
1. **Identify the Range**: The three-digit numbers we can form must be between 100 and 999.
2. **First Digit Choices**: Similar to the two-digit case, the first digit cannot be 0. Therefore, the possible choices for the first digit are again 1, 2, 3, 7, or 8. This gives us **5 options**.
3. **Second and Third Digit Choices**: Both the second and third digits can be any of the available digits (0, 1, 2, 3, 7, 8). This gives us **6 options** for each of these digits.
4. **Total Three-Digit Numbers**: The total number of three-digit numbers can be calculated by multiplying the number of choices for each digit:
\[ \text{Total Three-Digit Numbers} = 5 \times 6 \times 6 = 180 \]

### Step 3: Calculate the Total Numbers
1. **Combine the Results**: Now, we add the total number of two-digit numbers and three-digit numbers together:
\[ \text{Total Numbers} = \text{Total Two-Digit Numbers} + \text{Total Three-Digit Numbers} = 30 + 180 = 210 \]

### Final Answer
The total number of numbers that can be formed between 9 and 1000 using the digits 0, 1, 2, 3, 7, and 8 is **210**.

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