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Solution: Solve the system of equations. From the second equation, $x = 2y - 3$. Substitute into the first equation:
Question: Compute the square of $(2m - 3n)$ where $m$ and $n$ represent sensor readings in a control system.
We are given 10 distinguishable river sites, each to be assigned one of 3 pollution levels: high, moderate, or low. Since the order of classification does not matter, we are distributing 10 distinguishable objects into 3 distinguishable categories (levels), where each object belongs to exactly one category.
This is a classic problem of counting the number of functions from a set of size 10 (sites) to a set of size 3 (pollution levels). Each site can independently be assigned one of 3 labels.
Thus, the total number of ways to classify the sites is:
Therefore, the number of different classification schemes is:
Question: A paleontologist is excavating 6 distinct fossil sites and plans to assign each site to one of 3 research teams. If the teams are indistinguishable (i.e., only the grouping matters, not which team is which), how many ways can the 6 sites be assigned to the teams?
From known values of Stirling numbers:
Therefore, the number of ways to assign the 6 sites to indistinguishable teams is:
Question: A cartographer is labeling 8 different geographic features on a map using 3 types of labels: mountain, river, and valley. Each feature must receive exactly one label, and the cartographer wants to ensure that each label is used at least once. How many valid labelings are possible?
