Find the three-digit odd numbers that can be formed by using the digits 1, 2, 3, 4, 5, 6 when the repetition is allowed.

Explanation

To find the three-digit odd numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 with repetition allowed, we can follow these steps:

### Step 1: Identify the digits
The available digits are: 1, 2, 3, 4, 5, 6.

### Step 2: Determine the last digit
Since we want to form an odd number, the last digit must be odd. The odd digits available from our set are: 1, 3, and 5. Therefore, we have 3 choices for the last digit.

### Step 3: Determine the first digit
The first digit of a three-digit number can be any of the available digits (1, 2, 3, 4, 5, 6). Since repetition is allowed, we have 6 choices for the first digit.

### Step 4: Determine the second digit
Similarly, the second digit can also be any of the available digits (1, 2, 3, 4, 5, 6). Again, since repetition is allowed, we have 6 choices for the second digit.

### Step 5: Calculate the total number of combinations
To find the total number of three-digit odd numbers, we multiply the number of choices for each digit:
- Choices for the first digit: 6
- Choices for the second digit: 6
- Choices for the last digit: 3

Thus, the total number of three-digit odd numbers is:
\[ \text{Total} = (\text{Choices for first digit}) \times (\text{Choices for second digit}) \times (\text{Choices for last digit}) = 6 \times 6 \times 3 \]

### Step 6: Perform the calculation
Calculating the above expression:
\[ 6 \times 6 = 36 \]
\[ 36 \times 3 = 108 \]

### Final Answer
Therefore, the total number of three-digit odd numbers that can be formed is **108**.

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