Number of 7 digit telephone numbers that can be formed from the digits 0 , 1 , 2 , ,………………..9 , if each telephone number begins with digit 5 is

Options

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

Answer

To solve the problem of finding the number of 7-digit telephone numbers that can be formed from the digits 0 to 9, with the condition that each telephone number begins with the digit 5, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure of the Telephone Number**:
- A 7-digit telephone number is represented as follows:
\[ 5 \_ \_ \_ \_ \_ \_ \]
- The first digit is fixed as 5, leaving us with 6 remaining positions to fill.

2. **Identifying Available Digits**:
- The digits available for use are from 0 to 9, which gives us a total of 10 digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

3. **Filling the Remaining Positions**:
- Since there are no restrictions mentioned regarding repetition of digits, each of the 6 remaining positions can be filled with any of the 10 digits.
- Therefore, for each of the 6 positions, there are 10 choices.

4. **Calculating the Total Combinations**:
- The total number of combinations can be calculated by multiplying the number of choices for each of the 6 positions:
\[ 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^6 \]

5. **Final Result**:
- Thus, the total number of 7-digit telephone numbers that can be formed under the given conditions is:
\[ 10^6 \]

### Conclusion:
The answer is \( 10^6 \).

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