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But if the "order" refers only to the application of the operation (i.e., 2 positions are distinct for pumping), the count is $ \binom{7}{4} \cdot \binom{4}{2} \cdot \frac{4!}{2!} = 35 \cdot 6 \cdot 12 = 2520 $. Clarifying the problem's intent, the most plausible interpretation leads to $\boxed{2520}$.
Question: A nanotechnology engineer is designing a solar-powered water harvesting system with 5 identical microfilters. How many distinct ways can 8 unique water source samples be distributed into the filters if each filter can hold any number of samples?
Solution: Since the filters are identical and samples are distinguishable, this is equivalent to partitioning 8 distinct elements into up to 5 indistinct subsets. The total number of ways is the sum of Stirling numbers of the second kind for $ k = 1 $ to $ 5 $:

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