Question:** An environmental engineer is evaluating 6 different renewable energy sources. In how many ways can she select a subset of at least 2 and at most 5 sources to pilot in a new solar-wind hybrid system?

["Title: How Many Ways Can an Environmental Engineer Select a Subset of Renewable Energy Sources?\nMeta Description: A detailed breakdown of how many ways an environmental engineer can choose 2 to 5 out of 6 renewable energy sources for a solar-wind hybrid pilot project.", "---", "### Introduction", "When evaluating renewable energy options, environmental engineers often need to select subsets of technologies for real-world testing. In the case of developing a solar-wind hybrid system, a key question arises: In how many ways can the engineer select a subset of at least 2 and at most 5 renewable energy sources from a total of 6 available options?", "Whether you're a sustainability researcher, policy advisor, or project coordinator, understanding the combinations of energy sources empowers better planning and resource allocation. In this article, we explore the mathematical approach to determining how many meaningful subsets exist under these constraints.", "---", "### Understanding the Problem", "The engineer has 6 distinct renewable energy sources—say solar, wind, hydro, geothermal, biomass, and tidal—to choose from. She wants to test combinations using between 2 and 5 sources at a time.", "This is a classic combinatorics question involving combinations—specifically, selecting subsets of fixed sizes within a defined range.", "Not included:\n- Subsets of 0 sources (the empty set)\n- Subsets of 1 source\n- Subsets with 6 sources", "Required:\nSubsets with ≥ 2 sources and ≤ 5 sources.", "---", "### Mathematical Approach", "The number of ways to choose k elements from n distinct items is given by the binomial coefficient:", "[\n\binom{n}{k} = \frac{n!}{k!(n-k)!}\n]", "Here, n = 6 (total energy sources), k ranges from 2 to 5 inclusive.", "So, the total number of valid subsets is:", "[\n\binom{6}{2} + \binom{6}{3} + \binom{6}{4} + \binom{6}{5}\n]", "---", "### Step-by-Step Calculation", "Let’s compute each term:", "- (\binom{6}{2})\n [\n \frac{6!}{2! \cdot 4!} = \frac{6 \ imes 5}{2 \ imes 1} = 15\n ]", "- (\binom{6}{3})\n [\n \frac{6!}{3! \cdot 3!} = \frac{6 \ imes 5 \ imes 4}{3 \ imes 2 \ imes 1} = 20\n ]", "- (\binom{6}{4})\n [\n \binom{6}{4} = \binom{6}{2} = 15 \quad \ ext{(by symmetry of binomial coefficients)}\n ]", "- (\binom{6}{5})\n [\n \binom{6}{5} = \binom{6}{1} = 6\n ]", "---", "### Summing the Combinations", "Add all valid subset counts:", "[\n15 + 20 + 15 + 6 = 56\n]", "---", "### Conclusion", "There are 56 distinct ways an environmental engineer can select a subset of at least 2 and at most 5 renewable energy sources from 6 options to pilot in a solar-wind hybrid system.", "This calculation supports strategic decision-making in renewable energy research, ensuring engineers consider the full range of viable combinations without overextending resources with overly large or small groups.", "---", "### SEO Keywords for Optimization", "- Renewable energy combinations\n- Environmental engineering combinatorics\n- Subset selection for solar-wind hybrid systems\n- How many ways to choose renewable sources\n- Binomial coefficient applications\n- Energy pilot project planning", "---", "Ready to explore combinations for your next clean energy initiative? Start by calculating your ideal subset—and turn data into impact."]