The number of 4 digit even numbers that can be formed using 0, 1, 2, 3, 4, 5, 6 without repetition is

Options

A. 120
B. 300
C. 420
D. 20

Answer

To find the number of 4-digit even numbers that can be formed using the digits 0, 1, 2, 3, 4, 5, and 6 without repetition, we can break the problem down into cases based on the last digit, which must be even. The even digits available are 0, 2, 4, and 6.

### Step-by-Step Solution:

1. **Identify the last digit options**: The last digit of the 4-digit number must be even. The possible choices for the last digit are 0, 2, 4, and 6.

2. **Case 1: Last digit is 0**
- If the last digit is 0, the first digit can be any of the remaining digits: 1, 2, 3, 4, 5, or 6 (6 choices).
- After choosing the first digit, we have 5 digits left for the second position and 4 digits left for the third position.
- Therefore, the number of combinations in this case is:
\[ 6 \times 5 \times 4 = 120 \]

3. **Case 2: Last digit is 2**
- If the last digit is 2, the first digit can be any of the remaining digits except 0 (1, 3, 4, 5, or 6), which gives us 5 choices.
- The second digit can be any of the remaining 5 digits (including 0), and the third digit can be any of the remaining 4 digits.
- Therefore, the number of combinations in this case is:
\[ 5 \times 5 \times 4 = 100 \]

4. **Case 3: Last digit is 4**
- If the last digit is 4, the first digit can be any of the remaining digits except 0 (1, 2, 3, 5, or 6), which gives us 5 choices.
- The second digit can be any of the remaining 5 digits (including 0), and the third digit can be any of the remaining 4 digits.
- Therefore, the number of combinations in this case is:
\[ 5 \times 5 \times 4 = 100 \]

5. **Case 4: Last digit is 6**
- If the last digit is 6, the first digit can be any of the remaining digits except 0 (1, 2, 3, 4, or 5), which gives us 5 choices.
- The second digit can be any of the remaining 5 digits (including 0), and the third digit can be any of the remaining 4 digits.
- Therefore, the number of combinations in this case is:
\[ 5 \times 5 \times 4 = 100 \]

6. **Total Calculation**:
- Now, we sum the results from all cases:
\[ 120 + 100 + 100 + 100 = 420 \]

Thus, the total number of 4-digit even numbers that can be formed using the digits 0, 1, 2, 3, 4, 5, and 6 without repetition is **420**.