How many even numbers greater than 300 can be formed by the digits 1,2,3,4,5 no digit being repeated ?

Options

A. 111
B. 600
C. 900
D. none of these

Answer

To solve the problem of finding how many even numbers greater than 300 can be formed using the digits 1, 2, 3, 4, and 5 without repeating any digit, we can break it down into a step-by-step approach.

### Step 1: Identify the conditions
We need to form even numbers greater than 300 using the digits 1, 2, 3, 4, and 5. The even digits available are 2 and 4. Therefore, the last digit of our number must be either 2 or 4.

### Step 2: Consider the last digit as 2
If the last digit is 2, then the number must be greater than 300. The first digit can be 3, 4, or 5 (since 1 and 2 would make the number less than 300).

#### Case 1: Last digit is 2
- **First digit options**: 3, 4, 5 (3 options)
- **Middle digit options**: After choosing the first digit and the last digit (2), we have 3 digits left to choose from.

For each choice of the first digit, we can choose any of the remaining 3 digits for the middle position. Therefore, the number of combinations for this case is:
- Choices for the first digit: 3
- Choices for the middle digit: 3

Total combinations when last digit is 2:
\[ 3 \text{ (first digit)} \times 3 \text{ (middle digit)} = 9 \]

### Step 3: Consider the last digit as 4
If the last digit is 4, the first digit can again be 3, 2, or 5 (1 cannot be used as it would make the number less than 300).

#### Case 2: Last digit is 4
- **First digit options**: 3, 5 (2 options, since 2 cannot be used as the first digit)
- **Middle digit options**: After choosing the first digit and the last digit (4), we have 3 digits left to choose from.

For each choice of the first digit, we can choose any of the remaining 3 digits for the middle position. Therefore, the number of combinations for this case is:
- Choices for the first digit: 2
- Choices for the middle digit: 3

Total combinations when last digit is 4:
\[ 2 \text{ (first digit)} \times 3 \text{ (middle digit)} = 6 \]

### Step 4: Combine both cases
Now, we sum the total combinations from both cases:
- Total from Case 1 (last digit 2): 9
- Total from Case 2 (last digit 4): 6

Total even numbers greater than 300:
\[ 9 + 6 = 15 \]

### Conclusion
The total number of even numbers greater than 300 that can be formed using the digits 1, 2, 3, 4, and 5 without repetition is **15**.

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