Solution: This is a surjective function problem: count the number of ways to partition 7 distinct species into 4 non-empty habitats. Using the inclusion-exclusion principle: $ \sum_k=0^4 (-1)^k \binom4k (4 - k)^7 $. Calculating: $4^7 - \binom413^7 + \binom422^7 - \binom431^7$. Compute each term: $16384 - 4 \times 2187 + 6 \times 128 - 4 \times 1 = 16384 - 8748 + 768 - 4 = 8400$. The final answer is $\boxed8400$.**Question: - Groen Casting
Mar 01, 2026
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