Step 1: Understanding the Concept:
A number is divisible by 25 if and only if its last two digits are divisible by 25. From the given set $\{1, 2, 3, 4, 5, 6, 7\}$, we identify the possible two-digit endings.
Step 2: Key Formula or Approach:
The possible endings divisible by 25 are 25, 50, 75, and 00. Since we cannot use '0' and there is no repetition, only the endings $\{25\}$ and $\{75\}$ are valid.
Step 3: Detailed Explanation:
1. Case 1: Ending with '25'.
Last two digits are fixed. We need to choose 2 more digits for the thousands and hundreds places from the remaining 5 digits $\{1, 3, 4, 6, 7\}$.
Number of ways $= P(5, 2) = 5 \times 4 = 20$.
2. Case 2: Ending with '75'.
Last two digits are fixed. We need to choose 2 more digits from the remaining 5 digits $\{1, 2, 3, 4, 6\}$.
Number of ways $= P(5, 2) = 5 \times 4 = 20$.
Total numbers divisible by $25 = 20 + 20 = 40$.
Step 4: Final Answer:
There are 40 such numbers.