Solution: The total number of ways to select 5 startups is $\binom{9}{5}$. The favorable case is $\binom{4}{3} \cdot \binom{5}{2}$ (3 solar and 2 non-solar). Thus, the probability is $\frac{4 \cdot 10}{126} = \frac{40}{126} = \frac{20}{63}$. \boxed{\dfrac{20}{63}}### Question 1

["Understanding Combinatorial Selection: A Deep Dive into Startup Selection Probabilities", "When analyzing startup investments or research projects, a common task involves calculating how many ways to select a subset from a larger group — and then finding the probability of a specific favorable combination. This article breaks down one such precise scenario using combinatorial mathematics, focusing on the binomial coefficient (\binom{n}{k}), favorable outcomes, and probability computation.", "---", "### The Core Combinatorial Problem", "Suppose an investor or analyst wants to select 5 startups from a pool of 9, among which 4 focus on solar energy and 5 on other renewable or tech sectors. The mathematical model uses binomial coefficients to count:", "- Total ways to choose any 5 startups:\n [\n \binom{9}{5} = 126\n ]\n This represents all possible selections regardless of sector type.", "- Favorable selections: specifically, exactly 3 solar startups and 2 non-solar startups. To count this:\n - Choose 3 out of 4 solar startups: (\binom{4}{3} = 4)\n - Choose 2 out of 5 non-solar startups: (\binom{5}{2} = 10)\n - Multiply these:\n [\n \binom{4}{3} \cdot \binom{5}{2} = 4 \cdot 10 = 40\n ]\n So, there are 40 favorable combinations matching the desired sector split.", "---", "### Computing the Probability", "Probability is calculated as:\n[\n\ ext{Probability} = \frac{\ ext{Number of favorable outcomes}}{\ ext{Total number of outcomes}} = \frac{40}{126}\n]\nSimplifying the fraction:\n[\n\frac{40}{126} = \frac{20}{63}\n]\nThus, the probability of selecting exactly 3 solar startups and 2 non-solar startups when choosing 5 from this group is (\boxed{\dfrac{20}{63}}).", "---", "### Why This Matters", "This combinatorial approach is not just theoretical — it underpins real-world risk assessment, portfolio diversification strategies, and statistical inference in startup ecosystems. Understanding how to count and weight possible outcomes enables data-driven decisions with clear probabilistic foundations.", "Whether evaluating investment portfolios or analyzing sector distribution in innovation hubs, tools like (\binom{n}{k}) provide the rigor needed to move beyond intuition and toward measurable certainty.", "---", "Summary:\n- Total combinations: (\binom{9}{5} = 126)\n- Favorable outcomes (3 solar, 2 non-solar): (\binom{4}{3} \cdot \binom{5}{2} = 40)\n- Final probability: (\dfrac{20}{63})", "This structured methodology ensures clarity and correctness in probabilistic reasoning — a key skill for quantitative investors, researchers, and business strategists alike."]