The number of even numbers of four digits that can be formed using the digits 0, 1, 2, 3, 4 and 5 is

Options

A. 180
B. 156
C. 144
D. 198

Answer

To solve the problem of finding the number of even four-digit numbers that can be formed using the digits 0, 1, 2, 3, 4, and 5, we will break it down into cases based on the last digit, which must be even. The even digits available are 0, 2, and 4.

### Step 1: Case when the last digit is 0
- If the last digit is 0, the first digit cannot be 0 (to ensure it's a four-digit number). Therefore, the first digit can be chosen from {1, 2, 3, 4, 5} (5 options).
- After choosing the first digit, we have 4 remaining digits (including 0) to fill the second and third positions.
- We can choose the second digit from the remaining 4 digits and the third digit from the remaining 3 digits.

Calculating the number of combinations:
- First digit: 5 options
- Second digit: 4 options
- Third digit: 3 options

Total for this case:
\[ 5 \times 4 \times 3 = 60 \]

### Step 2: Case when the last digit is 2
- If the last digit is 2, the first digit can be chosen from {1, 3, 4, 5} (4 options, since 0 cannot be the first digit).
- After choosing the first digit, we have 4 remaining digits (including 0) to fill the second and third positions.
- We can choose the second digit from the remaining 4 digits and the third digit from the remaining 3 digits.

Calculating the number of combinations:
- First digit: 4 options
- Second digit: 4 options
- Third digit: 3 options

Total for this case:
\[ 4 \times 4 \times 3 = 48 \]

### Step 3: Case when the last digit is 4
- If the last digit is 4, the first digit can again be chosen from {1, 2, 3, 5} (4 options).
- After choosing the first digit, we have 4 remaining digits (including 0) to fill the second and third positions.
- We can choose the second digit from the remaining 4 digits and the third digit from the remaining 3 digits.

Calculating the number of combinations:
- First digit: 4 options
- Second digit: 4 options
- Third digit: 3 options

Total for this case:
\[ 4 \times 4 \times 3 = 48 \]

### Final Calculation
Now, we sum the totals from all three cases:
\[ 60 \text{ (last digit 0)} + 48 \text{ (last digit 2)} + 48 \text{ (last digit 4)} = 156 \]

Thus, the total number of even four-digit numbers that can be formed using the digits 0, 1, 2, 3, 4, and 5 is **156**.

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