Key Concepts and Formulas

• Fundamental Principle of Counting (Multiplication Principle): If there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \times n$ ways to do both.
• Definition of an Odd Number: An integer is odd if its units digit is odd (i.e., 1, 3, 5, 7, 9, etc.).
• Definition of a Four-Digit Number: A number with four digits, where the leading digit (thousands place) cannot be zero.

Step-by-Step Solution

We want to find the number of four-digit odd numbers that can be formed using the digits {0, 1, 2, 3, 5, 7}, with repetition allowed. Let the four-digit number be represented as $\underline{D_1} \ \underline{D_2} \ \underline{D_3} \ \underline{D_4}$.

Step 1: Filling the Units Place ($D_4$)

• What & Why: We need to ensure the number is odd. The units digit ($D_4$) must be an odd number. We fill this position first because it is the most restrictive condition.
• Available Digits: From the set {0, 1, 2, 3, 5, 7}, the odd digits are 1, 3, 5, and 7.
• Number of Choices: There are 4 choices for the units place. $\underline{\hspace{0.5cm}} \ \underline{\hspace{0.5cm}} \ \underline{\hspace{0.5cm}} \ \underline{4}$

Step 2: Filling the Thousands Place ($D_1$)

• What & Why: We need to ensure the number is a four-digit number. The thousands digit ($D_1$) cannot be 0. We fill this position next as it has the second most restrictive condition.
• Available Digits: From the set {0, 1, 2, 3, 5, 7}, the digits that are not 0 are 1, 2, 3, 5, and 7.
• Number of Choices: There are 5 choices for the thousands place. $\underline{5} \ \underline{\hspace{0.5cm}} \ \underline{\hspace{0.5cm}} \ \underline{4}$

Step 3: Filling the Hundreds Place ($D_2$)

• What & Why: There are no specific restrictions for the hundreds place ($D_2$) other than the digit must come from the given set. Repetition is allowed.
• Available Digits: All digits from the set {0, 1, 2, 3, 5, 7} are available.
• Number of Choices: There are 6 choices for the hundreds place. $\underline{5} \ \underline{6} \ \underline{\hspace{0.5cm}} \ \underline{4}$

Step 4: Filling the Tens Place ($D_3$)

• What & Why: Similar to the hundreds place, no specific restrictions apply for the tens place ($D_3$). Repetition is allowed.
• Available Digits: All digits from the set {0, 1, 2, 3, 5, 7} are available.
• Number of Choices: There are 6 choices for the tens place. $\underline{5} \ \underline{6} \ \underline{6} \ \underline{4}$

Step 5: Applying the Fundamental Principle of Counting

• What & Why: To find the total number of possible four-digit odd numbers, we multiply the number of choices for each position, according to the Fundamental Principle of Counting.
• Calculation: $\text{Total Number} = (\text{Choices for } D_1) \times (\text{Choices for } D_2) \times (\text{Choices for } D_3) \times (\text{Choices for } D_4)$ $\text{Total Number} = 5 \times 6 \times 6 \times 4$ $\text{Total Number} = 30 \times 24$ $\text{Total Number} = 720$

Common Mistakes & Tips

• Zero in the First Place: Always remember that zero cannot be the first digit of a multi-digit number.
• Repetition Allowed: If repetition is allowed, the number of choices for each position remains constant unless there are other restrictions.
• Prioritize Restrictions: Fill the positions with the most restrictions first.

Summary

By systematically filling each digit position according to the given constraints and allowing for repetition, we calculated the total number of possible four-digit odd numbers. We addressed the most restrictive conditions (units digit for oddness, thousands digit for being a four-digit number) first, and then filled the remaining positions with all available choices due to repetition being allowed. This method led to 720 such numbers.

The final answer is \boxed{720}, which corresponds to option (D).