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. 100000
C. 6720
D. 30240

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 right is 5 and the digit at the extreme left is 6, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Fixed Digits**:
- The digit at the extreme left (first position) is fixed as 6.
- The digit at the extreme right (last position) is fixed as 5.

2. **Determine Remaining Digits**:
- Since we are using the digits 0 to 9, we have a total of 10 digits.
- With 6 and 5 already used, we have 8 digits left to choose from: 0, 1, 2, 3, 4, 7, 8, 9.

3. **Calculate the Number of Ways to Fill the Remaining Positions**:
- The remaining positions to fill are the 5 middle digits (positions 2, 3, 4, 5, and 6).
- We need to select 5 digits from the 8 available digits.

4. **Choose and Arrange the Digits**:
- The number of ways to select and arrange 5 digits from the 8 available digits can be calculated as follows:
- The first middle position can be filled by any of the 8 remaining digits.
- The second middle position can be filled by any of the 7 remaining digits (after filling the first).
- The third middle position can be filled by any of the 6 remaining digits.
- The fourth middle position can be filled by any of the 5 remaining digits.
- The fifth middle position can be filled by any of the 4 remaining digits.

5. **Calculate the Total Combinations**:
- The total number of combinations can be calculated as:
\[ \text{Total Combinations} = 8 \times 7 \times 6 \times 5 \times 4 \]

6. **Perform the Calculation**:
- Now, we can compute the product:
\[ 8 \times 7 = 56 \]
\[ 56 \times 6 = 336 \]
\[ 336 \times 5 = 1680 \]
\[ 1680 \times 4 = 6720 \]

7. **Final Result**:
- Therefore, the total number of different 7-digit telephone numbers possible under the given conditions is **6720**.