How many 5-digits telephone numbers can be formed with the digits 0,1,2,….,8,9 if each numbers starts with 35 and no digit appears more than once?

Answer

To solve the problem of how many 5-digit telephone numbers can be formed starting with "35" and using the digits 0 to 9 without repeating any digit, we can follow these steps:

### Step 1: Identify the fixed digits
The telephone number must start with "35". Therefore, the first two digits are fixed:
- First digit: 3
- Second digit: 5

### Step 2: Determine the available digits
Since we are using the digits from 0 to 9, the available digits are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9

However, since 3 and 5 are already used, we cannot use them again. This leaves us with the following digits to choose from:
0, 1, 2, 4, 6, 7, 8, 9 (a total of 8 digits).

### Step 3: Choose the remaining digits
We need to fill the remaining 3 positions in the telephone number. The available digits are 0, 1, 2, 4, 6, 7, 8, 9.

1. For the **third digit**, we can choose any of the 8 remaining digits.
2. For the **fourth digit**, we can choose from the remaining 7 digits (since one digit has already been used).
3. For the **fifth digit**, we can choose from the remaining 6 digits (since two digits have already been used).

### Step 4: Calculate the total combinations
The total number of ways to fill the last three positions can be calculated by multiplying the number of choices for each position:
- Choices for the third digit: 8
- Choices for the fourth digit: 7
- Choices for the fifth digit: 6

Thus, the total number of 5-digit telephone numbers is:
\[ 8 \times 7 \times 6 = 336 \]

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

---