How many 7 digit telephone numbers can be formed from the digits 0, 1, 2, ..., 9, where each telephone number begins with digit 2

Options

A. `10^6`
B. `6^(10)`
C. `10!`
D. `""^(10)P_6`

Answer

To solve the problem of how many 7-digit telephone numbers can be formed from the digits 0 to 9, where each telephone number begins with the digit 2, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Structure of the Telephone Number**:
- A 7-digit telephone number has the format: 2XXXXXX, where 'X' represents any digit from 0 to 9.

2. **Determine the Fixed Digit**:
- The first digit is fixed as '2'. This means we only need to determine the remaining 6 digits.

3. **Count the Possible Digits for Each Position**:
- For each of the remaining 6 positions (XXXXXX), we can use any digit from 0 to 9. This gives us a total of 10 possible choices for each of these positions.

4. **Calculate the Total Combinations**:
- Since each of the 6 positions can be filled independently with any of the 10 digits, we can calculate the total number of combinations using the formula:
\[ \text{Total Combinations} = 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^6 \]

5. **Final Answer**:
- Therefore, the total number of 7-digit telephone numbers that can be formed is \(10^6\).

### Conclusion:
The total number of 7-digit telephone numbers that can be formed, starting with the digit 2, is \(10^6\).

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