How many 5 digits telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?

Explanation

To solve the problem of how many 5-digit telephone numbers can be constructed using the digits 0 to 9, starting with 67 and ensuring no digit appears more than once, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Fixed Digits**:
The telephone number must start with the digits 6 and 7. Therefore, the first two digits of the telephone number are fixed as 6 and 7.

2. **Count Remaining Digits**:
After using the digits 6 and 7, we have the following digits left to choose from: 0, 1, 2, 3, 4, 5, 8, 9. This gives us a total of 8 available digits.

3. **Determine the Number of Digits to Fill**:
Since we have already used 2 digits (6 and 7), we need to fill 3 more digits to complete the 5-digit telephone number.

4. **Choose the Remaining Digits**:
We can select the remaining 3 digits from the 8 available digits. The number of ways to choose and arrange these 3 digits can be calculated using permutations since the order matters.

5. **Calculate the Permutations**:
The number of ways to choose and arrange 3 digits from 8 can be calculated as follows:
- For the first digit, we have 8 options.
- For the second digit, we have 7 options left (since one digit has already been used).
- For the third digit, we have 6 options left.

Therefore, the total number of arrangements can be calculated as:
\[ \text{Total arrangements} = 8 \times 7 \times 6 \]

6. **Perform the Calculation**:
\[ 8 \times 7 = 56 \]
\[ 56 \times 6 = 336 \]

7. **Final Count of Telephone Numbers**:
Since the first two digits are fixed (67), the total number of different 5-digit telephone numbers that can be formed is 336.

### Final Answer:
The total number of 5-digit telephone numbers that can be constructed is **336**.