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 will break the problem into two cases based on the last digit (which must be even).

### Step 1: Identify the even digits
The even digits available from the set {0, 1, 2, 3, 4, 5, 6} are 0, 2, 4, and 6.

### Step 2: Case 1 - Last digit is 0
If the last digit (d4) is 0, we can choose the first three digits (d1, d2, d3) from the remaining digits {1, 2, 3, 4, 5, 6}.

- **Choosing d1:** We can choose d1 from 1, 2, 3, 4, 5, 6 (6 options).
- **Choosing d2:** After choosing d1, we have 5 digits left to choose from.
- **Choosing d3:** After choosing d2, we have 4 digits left to choose from.

Thus, the total arrangements for this case are:
\[ \text{Total for Case 1} = 6 \times 5 \times 4 = 120 \]

### Step 3: Case 2 - Last digit is not 0
If the last digit (d4) is one of the other even digits (2, 4, or 6), we can choose d4 in 3 ways (2, 4, or 6).

- **Choosing d1:** d1 cannot be 0 and cannot be equal to d4. So, we can choose d1 from the remaining digits {1, 3, 5} plus the two other even digits that are not chosen as d4. This gives us 5 options for d1.
- **Choosing d2:** After choosing d1 and d4, we have 5 digits left.
- **Choosing d3:** After choosing d2, we have 4 digits left.

Thus, the total arrangements for this case are:
\[ \text{Total for Case 2} = 3 \times 5 \times 5 \times 4 = 300 \]

### Step 4: Combine the results
Now, we combine the results from both cases:
\[ \text{Total even 4-digit numbers} = \text{Total for Case 1} + \text{Total for Case 2} = 120 + 300 = 420 \]

### Final Answer
The total number of 4-digit even numbers that can be formed is **420**.

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