There is a 7-digit telephone number with all different digits. If the digit at extreme right and extreme left are 5 and 6 respectively, find how many such telephone numbers are possible.

Options

A. 120
B. 1,00,000
C. 6720
D. None of these

Answer

To solve the problem of finding how many 7-digit telephone numbers can be formed with all different digits, where the digit at the extreme left is 6 and the digit at the extreme right is 5, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Fixed Digits**:
- The extreme left digit is 6.
- The extreme right digit is 5.
- Therefore, the telephone number format is: 6 _ _ _ _ 5.

2. **List the Available Digits**:
- The digits available for the remaining positions are: 0, 1, 2, 3, 4, 7, 8, 9.
- Since we have already used 5 and 6, we are left with 8 digits.

3. **Determine the Number of Positions to Fill**:
- There are 5 positions left to fill (the second to the sixth positions in the number).

4. **Calculate the Number of Choices for Each Position**:
- For the second position, we can choose any of the 8 available digits.
- After choosing a digit for the second position, we have 7 digits left for the third position.
- After choosing a digit for the third position, we have 6 digits left for the fourth position.
- After choosing a digit for the fourth position, we have 5 digits left for the fifth position.
- After choosing a digit for the fifth position, we have 4 digits left for the sixth position.

5. **Multiply the Choices**:
- The total number of ways to fill the 5 positions is calculated as follows:
\[ \text{Total Ways} = 8 \times 7 \times 6 \times 5 \times 4 \]

6. **Perform the Calculation**:
- Calculate step by step:
- \( 8 \times 7 = 56 \)
- \( 56 \times 6 = 336 \)
- \( 336 \times 5 = 1680 \)
- \( 1680 \times 4 = 6720 \)

7. **Final Result**:
- Therefore, the total number of different 7-digit telephone numbers that can be formed is **6720**.