How many 6 digit telephone numbers can be constructed with the digits 0,1,2,3,4,5,6,7,8,9 if each number starts with 35 and no digit appears more than once ?

Options

A. 1600
B. 1680
C. 900000
D. 9000

Answer

To find the number of 6-digit telephone numbers that can be constructed starting with the digits 35, and ensuring that no digit appears more than once, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Fixed Digits**:
The first two digits of the telephone number are fixed as 3 and 5. Therefore, we have:
- First digit: 3
- Second digit: 5

2. **Determine Remaining Digits**:
The available digits are from 0 to 9, which gives us the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Since 3 and 5 are already used, the remaining digits we can use are:
- Remaining digits: {0, 1, 2, 4, 6, 7, 8, 9}
- Total remaining digits = 8 (since we have used 2 digits)

3. **Choose the Remaining Digits**:
We need to fill the remaining 4 positions (3rd, 4th, 5th, and 6th) in the telephone number. The choices for these positions will be from the 8 remaining digits.

4. **Calculate the Number of Choices**:
- For the 3rd digit, we can choose any of the 8 remaining digits.
- For the 4th digit, we can choose from the remaining 7 digits (since one digit has already been used).
- For the 5th digit, we can choose from the remaining 6 digits.
- For the 6th digit, we can choose from the remaining 5 digits.

5. **Multiply the Choices**:
The total number of different combinations can be calculated by multiplying the number of choices for each digit:
\[ \text{Total combinations} = 8 \times 7 \times 6 \times 5 \]

6. **Calculate the Result**:
\[ 8 \times 7 = 56 \\ 56 \times 6 = 336 \\ 336 \times 5 = 1680 \]
Therefore, the total number of 6-digit telephone numbers that can be constructed is **1680**.

### Final Answer:
The total number of 6-digit telephone numbers that can be constructed starting with 35 and without repeating any digits is **1680**.