How many 3 digit even numbers are there such if 3 is one of the digits then 9 is the next digits ?

Options

A. 365
B. 536
C. 210
D. 156

Answer

To solve the problem of finding how many 3-digit even numbers there are such that if 3 is one of the digits, then 9 is the next digit, we can break it down into steps.

### Step-by-Step Solution

1. **Identify the last digit**:
Since we are looking for even numbers, the last digit must be one of the even digits: 0, 2, 4, 6, or 8. This gives us 5 possibilities for the last digit.

**Hint**: Remember that even numbers end with 0, 2, 4, 6, or 8.

2. **Determine the conditions for the digits**:
If 3 is one of the digits, then the next digit must be 9. This means we need to consider two cases:
- Case 1: The number contains the digits 3 and 9.
- Case 2: The number does not contain the digits 3 and 9.

3. **Case 1 - When 3 and 9 are present**:
- If 3 is in the number, it can be in either the hundreds or the tens place.
- If 3 is in the tens place, then the hundreds place can be any digit from 1 to 9 (except 3) and the last digit can be any even digit (0, 2, 4, 6, 8). This gives us:
- Hundreds place: 8 options (1-9 excluding 3)
- Tens place: 3
- Last digit: 5 options (0, 2, 4, 6, 8)
- If 3 is in the hundreds place, then the tens place must be 9, and the last digit can be any even digit. This gives us:
- Hundreds place: 1 option (3)
- Tens place: 1 option (9)
- Last digit: 5 options (0, 2, 4, 6, 8)

4. **Calculating possibilities for Case 1**:
- For 3 in the tens place: \(8 \times 1 \times 5 = 40\)
- For 3 in the hundreds place: \(1 \times 1 \times 5 = 5\)
- Total for Case 1: \(40 + 5 = 45\)

5. **Case 2 - When 3 and 9 are not present**:
- The hundreds place can be any digit from 1 to 9 (excluding 3), giving us 8 options.
- The tens place can be any digit from 0 to 9 (excluding 3 and 9), giving us 8 options.
- The last digit can still be any even digit (0, 2, 4, 6, 8), giving us 5 options.

6. **Calculating possibilities for Case 2**:
- Total for Case 2: \(8 \times 8 \times 5 = 320\)

7. **Final Calculation**:
- Total even numbers satisfying the conditions = Case 1 + Case 2 = \(45 + 320 = 365\)

### Conclusion
The total number of 3-digit even numbers such that if 3 is one of the digits, then 9 is the next digit is **365**.