An environmental scientist is analyzing data on water samples taken from different sites along the Everglades. The scientist has collected 5 water samples from Site A, 4 samples from Site B, and 3 samples from Site C. In how many distinct sequences can the scientist analyze the samples if samples from the same site are indistinguishable? - Sterling Industries

An environmental scientist is analyzing data on water samples taken from different sites along the Everglades. With ongoing efforts to protect one of North America’s most vital wetlands, monitoring water quality plays a crucial role in understanding ecosystem health. This interest has grown as researchers track changes in nutrient levels, contamination, and overall biodiversity across key zones. The recent study examines five samples from Site A, four from Site B, and three from Site C—data points essential for measuring environmental shifts over time.

Why is analyzing water samples from multiple Everglades sites so important? Water quality impacts wildlife, agriculture, drinking supplies, and restoration projects. By comparing samples from distinct locations, scientists detect localized pollution sources, seasonal trends, and the effectiveness of conservation measures. This approach ensures targeted, science-based decisions shape preservation strategies. With widespread digital engagement in environmental data, understanding how to manage and interpret these samples has become increasingly relevant across the US.

How Does an Environmental Scientist Analyze Synthetic Water Samples? When analyzing samples from the same site as indistinguishable, scientists use combinatorics to calculate distinct sequencing possibilities. The total number of ordered sequences is determined by the multinomial formula: n! divided by the product of factorials of identical site counts.
Here, 5 samples from Site A, 4 from Site B, and 3 from Site C make a total of 12 samples. The unique sequence counts equal 12! divided by (5! × 4! × 3!). This formula accounts for repeated elements — treating identical site samples as interchangeable — providing an accurate representation of possible analytical orderings without overcounting.