Key Concepts and Formulas

• Fundamental Principle of Counting (Multiplication Rule): 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.
• Casework: Breaking down a problem into mutually exclusive cases and summing the results.
• Understanding Inequalities: Interpreting "not more than" as "less than or equal to."

Step-by-Step Solution

Step 1: Analyze the constraints on the first digit ($d_1$).

The problem requires a 5-digit number greater than 50000. Therefore, the first digit, $d_1$, must be one of $\{5, 6, 7\}$. The available digits are $\{0, 1, 2, 3, 4, 5, 6, 7\}$. Thus, $d_1$ can be 5, 6, or 7.

Step 2: Analyze the constraint on the last digit ($d_5$) based on $d_1 + d_5 \le 8$.

We use casework based on the value of $d_1$.

• Case 1: $d_1 = 5$ The constraint $d_1 + d_5 \le 8$ becomes $5 + d_5 \le 8$. Subtracting 5 from both sides gives $d_5 \le 3$. The possible values for $d_5$ are $\{0, 1, 2, 3\}$. Thus, there are 4 choices for $d_5$.

• Case 2: $d_1 = 6$ The constraint $d_1 + d_5 \le 8$ becomes $6 + d_5 \le 8$. Subtracting 6 from both sides gives $d_5 \le 2$. The possible values for $d_5$ are $\{0, 1, 2\}$. Thus, there are 3 choices for $d_5$.

• Case 3: $d_1 = 7$ The constraint $d_1 + d_5 \le 8$ becomes $7 + d_5 \le 8$. Subtracting 7 from both sides gives $d_5 \le 1$. The possible values for $d_5$ are $\{0, 1\}$. Thus, there are 2 choices for $d_5$.

Step 3: Analyze the possible values for the middle digits ($d_2, d_3, d_4$).

The digits $d_2, d_3,$ and $d_4$ have no restrictions other than being chosen from the set $\{0, 1, 2, 3, 4, 5, 6, 7\}$. Since repetition is allowed, each of these digits can be any of the 8 digits. Therefore, there are 8 choices for each of $d_2, d_3,$ and $d_4$. The number of ways to choose these digits is $8 \times 8 \times 8 = 8^3 = 512$.

Step 4: Calculate the total number of possibilities for each case.

• Case 1: $d_1 = 5$ There is 1 choice for $d_1$ (5), 4 choices for $d_5$, and 512 choices for $d_2, d_3, d_4$. The total number of possibilities for this case is $1 \times 4 \times 512 = 2048$.

• Case 2: $d_1 = 6$ There is 1 choice for $d_1$ (6), 3 choices for $d_5$, and 512 choices for $d_2, d_3, d_4$. The total number of possibilities for this case is $1 \times 3 \times 512 = 1536$.

• Case 3: $d_1 = 7$ There is 1 choice for $d_1$ (7), 2 choices for $d_5$, and 512 choices for $d_2, d_3, d_4$. The total number of possibilities for this case is $1 \times 2 \times 512 = 1024$.

Step 5: Calculate the grand total.

Since the cases are mutually exclusive, we sum the number of possibilities for each case to find the total number of 5-digit numbers satisfying the given conditions.

Total number of such numbers = $2048 + 1536 + 1024 = 4608$.

Common Mistakes & Tips

• Assuming Distinct Digits: The problem does not state that the digits must be distinct. Always check this assumption before proceeding.
• Incorrectly Interpreting "Not More Than": "$A$ not more than $B$" means $A \le B$, not $A < B$.
• Missing a Case: Ensure that all possible values for $d_1$ are considered in the casework.

Summary

By analyzing the constraints, breaking the problem into cases based on the first digit, and applying the fundamental principle of counting, we found that there are 4608 such 5-digit numbers.

The final answer is $\boxed{4608}$, which corresponds to option (C).