Question: A sustainable agriculture startup divides 6 distinct crop types into 2 identical storage units. How many ways can the crops be partitioned if each unit must contain at least one type?
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- Question: A neural interface system processes 5 distinct signals using 3 unique channels. How many ways can the signals be assigned to channels if each channel can process any number of signals?
- Solution: Each of the 5 distinct signals has 3 independent choices (channels). By the multiplication principle, the total number of assignments is $3^5$. Calculating:
- The number of ways is $\boxed{243}$.
- Solution: This is equivalent to finding the number of ways to partition 6 distinct elements into 2 non-empty subsets, which is the Stirling number of the second kind $S(6, 2)$. The formula for $S(n, 2)$ is $2^{n-1} - 1$. For $n = 6$:
- 2^{5} - 1 = 32 - 1 = 31.
- The number of ways is $\boxed{31}$.
