Sam has forgotten his friend's seven-digit telephone number. He remembers the following: the first three digits are either 635 or 674, the number is odd, and the number nine appears once, what is the minimum number of trials he has to make before he can be certain to succeed?

Options

A. 1000
B. 2430
C. 3402
D. 3006

Answer

To solve the problem, we need to determine the minimum number of trials Sam has to make to be certain to find his friend's seven-digit telephone number based on the given conditions.

### Step-by-Step Solution:

1. **Identify the First Three Digits**:
- The first three digits of the telephone number can either be **635** or **674**. This gives us **2 options** for the first three digits.

2. **Determine the Conditions**:
- The number is **odd**. Therefore, the last digit must be one of the odd digits: **1, 3, 5, 7, or 9**.
- The digit **9** appears **exactly once** in the number.

3. **Case 1: First Three Digits = 635**:
- If the first three digits are **635**, we need to consider the placement of the digit **9** and the remaining digits.
- The last digit can be **1, 3, 5, or 7** (4 options) since **9** cannot be placed here (it would not be odd).
- The digit **9** can occupy one of the following positions: 4th, 5th, or 6th.

- **Sub-case 1: 9 in the 4th position**:
- Remaining digits (5th, 6th, and 7th) can be filled with any digits from **0 to 8** (9 options each).
- Last digit must be odd (1, 3, 5, 7) = 4 options.
- Total combinations: \(1 \times 9 \times 9 \times 4 = 324\).

- **Sub-case 2: 9 in the 5th position**:
- Remaining digits (4th, 6th, and 7th) can be filled with any digits from **0 to 8** (9 options each).
- Last digit must be odd (1, 3, 5, 7) = 4 options.
- Total combinations: \(9 \times 1 \times 9 \times 4 = 324\).

- **Sub-case 3: 9 in the 6th position**:
- Remaining digits (4th, 5th, and 7th) can be filled with any digits from **0 to 8** (9 options each).
- Last digit must be odd (1, 3, 5, 7) = 4 options.
- Total combinations: \(9 \times 9 \times 1 \times 4 = 324\).

- **Sub-case 4: 9 in the 7th position**:
- Remaining digits (4th, 5th, and 6th) can be filled with any digits from **0 to 8** (9 options each).
- Total combinations: \(9 \times 9 \times 9 \times 1 = 729\).

- **Total for 635**:
- \(324 + 324 + 324 + 729 = 1701\).

4. **Case 2: First Three Digits = 674**:
- The analysis is the same as for the first case.
- Total for 674:
- \(324 + 324 + 324 + 729 = 1701\).

5. **Final Calculation**:
- Total combinations for both cases:
- \(1701 + 1701 = 3402\).

### Conclusion:
The minimum number of trials Sam has to make before he can be certain to succeed in finding his friend's telephone number is **3402**.