Question:

To form 7-digit numbers using digits {1, 2, 3, 4, 5}, we need to use 7 positions with only 5 distinct digits available. By the Pigeonhole Principle, at least two digits must repeat. We use the inclusion-exclusion principle: Total arrangements = $5^7$ (each position can have any of 5 digits). We subtract cases where at most one digit appears, add back cases where at most two distinct digits appear, and so on. The number of surjective functions from 7 positions to 5 digits is $5! \cdot S(7,5)$ where $S(7,5)$ is Stirling number of second kind, giving $5^7 - inom{5}{1}4^7 + inom{5}{2}3^7 - inom{5}{3}2^7 + inom{5}{4}1^7 = 78125 - 20480 + 2430 - 80 + 5 = 60000$.