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.
• Permutation: An arrangement of objects in a specific order.
• Understanding the problem's constraints and how they limit the choices for each position is crucial.

Step-by-Step Solution

Step 1: Analyze the Constraints and Determine the Number of Digits

• What and Why: We need to determine the number of digits the numbers we are forming must have. The problem specifies that the numbers must be strictly between 5000 and 10000. This means they must be greater than 5000 and less than 10000.
• Explanation: Numbers between 5000 and 10000 must be 4-digit numbers. A 3-digit number is at most 999, which is less than 5000. A 5-digit number is at least 10000, violating the upper bound.
• Conclusion: We are forming 4-digit numbers.

Step 2: Identify Available Digits

• What and Why: We need to list the digits we can use to form the numbers. The problem states that we can only use the digits 1, 3, 5, 7, and 9.
• Available Digits: The set of available digits is $S = \{1, 3, 5, 7, 9\}$.

Step 3: Determine Choices for the Thousands Place

• What and Why: We start by considering the thousands place because the constraint "strictly between 5000 and 10000" most directly affects this digit.
• Explanation: The number must be greater than 5000, so the thousands digit must be 5, 7, or 9. If we use 1 or 3 in the thousands place, the number will be less than 5000.
• Calculation: The possible digits for the thousands place are {5, 7, 9}. Therefore, there are 3 choices.
• Result: Thousands place: 3 choices.

Step 4: Determine Choices for the Hundreds Place

• What and Why: Now we consider the hundreds place. We must account for the "no repetition" rule.
• Explanation: One digit has already been used in the thousands place. Since we have 5 digits in total, there are now $5 - 1 = 4$ digits remaining that can be used in the hundreds place.
• Calculation: There are 4 choices for the hundreds place.
• Result: Hundreds place: 4 choices.

Step 5: Determine Choices for the Tens Place

• What and Why: We move to the tens place, again considering the "no repetition" rule.
• Explanation: Two digits have already been used (one in the thousands place and one in the hundreds place). Therefore, there are $5 - 2 = 3$ digits remaining that can be used in the tens place.
• Calculation: There are 3 choices for the tens place.
• Result: Tens place: 3 choices.

Step 6: Determine Choices for the Units Place

• What and Why: We move to the units place, again considering the "no repetition" rule.
• Explanation: Three digits have already been used (in the thousands, hundreds, and tens places). Therefore, there are $5 - 3 = 2$ digits remaining that can be used in the units place.
• Calculation: There are 2 choices for the units place.
• Result: Units place: 2 choices.

Step 7: Calculate the Total Number of Possible Numbers

• What and Why: Using the Fundamental Principle of Counting, we multiply the number of choices for each place value to find the total number of possible numbers.
• Calculation: Total number of numbers = (Choices for Thousands) $\times$ (Choices for Hundreds) $\times$ (Choices for Tens) $\times$ (Choices for Units) $3 \times 4 \times 3 \times 2 = 72$

Common Mistakes & Tips

• Forgetting the "greater than 5000" constraint: This is the most common mistake. Remember to restrict the thousands digit to 5, 7, or 9.
• Ignoring the "no repetition" constraint: Each time you fill a place value, you reduce the number of available digits for the remaining places.

Summary

We determined that the numbers must be 4-digit numbers. We then considered the restrictions on each digit, particularly the thousands digit (must be 5, 7, or 9). Applying the Fundamental Principle of Counting, we found that there are $3 \times 4 \times 3 \times 2 = 72$ such numbers.

Final Answer

The final answer is \boxed{72}, which corresponds to option (C).