The number of ways is $\boxed{31}$.
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- 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?
- 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.
- Question: If three neurons each fire independently with a probability of $\frac{1}{4}$ per second, what is the probability that exactly one fires in a given second?
- Solution: Using the binomial distribution with $n = 3$, $p = \frac{1}{4}$, and $k = 1$:
- \binom{3}{1} \left(\frac{1}{4}\right)^1 \left(\frac{3}{4}\right)^2 = 3 \cdot \frac{1}{4} \cdot \frac{9}{16} = \frac{27}{64}.
