How many even numbers of three digits each can be made with the digits 1, 2, 3, 4, 5, 6 if : no digit is repeated.

Answer

To find how many even numbers of three digits can be formed using the digits 1, 2, 3, 4, 5, and 6 without repeating any digit, we can follow these steps:

### Step 1: Identify the possible last digits
Since we want to form even numbers, the last digit must be an even digit. The even digits available from the set {1, 2, 3, 4, 5, 6} are 2, 4, and 6. Thus, we have 3 choices for the last digit.

**Hint:** Think about the properties of even numbers and which digits qualify as even.

### Step 2: Choose the last digit
Let’s denote the last digit as \(d_3\). We can choose \(d_3\) from the even digits: 2, 4, or 6.

**Choices for \(d_3\):**
- If \(d_3 = 2\)
- If \(d_3 = 4\)
- If \(d_3 = 6\)

### Step 3: Choose the first and second digits
After selecting the last digit, we have 5 remaining digits to choose from for the first two positions (\(d_1\) and \(d_2\)).

For each choice of \(d_3\):
- We can choose \(d_1\) from the remaining 5 digits.
- After choosing \(d_1\), we will have 4 digits left to choose from for \(d_2\).

### Step 4: Calculate the total combinations
For each choice of the last digit, the number of ways to choose the first two digits is:
- Number of ways to choose \(d_1\) = 5 (remaining digits)
- Number of ways to choose \(d_2\) = 4 (after choosing \(d_1\))

Thus, the total number of combinations for each choice of \(d_3\) is:
\[ \text{Total combinations for each } d_3 = 5 \times 4 = 20 \]

Since there are 3 choices for \(d_3\), the total number of even three-digit numbers is:
\[ \text{Total even numbers} = 3 \times (5 \times 4) = 3 \times 20 = 60 \]

### Final Answer
Therefore, the total number of even three-digit numbers that can be formed is **60**.

---