#### 15Question: An angel investor decides to fund a startup every month, choosing one of three sectors: AI, biotech, or clean energy. If the investor randomly selects a sector each month with equal probability, what is the probability that over the next 6 months, she funds exactly two AI startups, two biotech startups, and two clean energy startups?

["##### 15Question: An angel investor funds a startup monthly across AI, biotech, and clean energy with equal odds—what’s the math behind achieving exactly two in each category over six months?", "In today’s fast-evolving investment landscape, a growing number of U.S. investors are tracking how early-stage capital flow shapes major technological and societal shifts. One fascinating question emerging from startup analytics isn’t about hype—but about statistical balance: if every month an angel investor randomly selects one of three sectors—AI, biotech, or clean energy—with full equal probability, what’s the chance she funds exactly two AI, two biotech, and two clean energy ventures in a 6-month window? This scenario reflects both the unpredictability of innovation funding and the data-driven approach reshaping venture investment strategies across the country.", "---", "### Why This Question Matters in the US Startup Ecosystem", "With venture capital increasingly concentrated in high-impact sectors like artificial intelligence and climate tech, understanding how randomness translates into real-world sector distribution has become a key analytical focus. Investors and market observers are interested in how a balanced monthly choice—each sector equally likely—creates near-equal representation across these categories. This balance matters not just for risk diversification but also for assessing long-term portfolio health and identifying emerging market momentum.", "The equal 1/3 probability structure makes this a clean combinatorics problem—ideal for educators, data explorers, and curious professionals seeking clarity in a complex market.", "---", "### How the Probability Works—Breaking It Down", "To fund exactly two AI, two biotech, and two clean energy startups over six months, the investor must make six independent, probabilistic selections across three equally likely categories. The challenge lies in counting all possible sequences of funding choices that meet this exact distribution and dividing by all possible combinations across six selections.", "Think of it as distributing six “funding slots” into three distinct boxes (sectors) with exactly two slots per box. This is a classic multinomial probability scenario—where outcomes are grouped into categories with fixed constraints.", "---", "### Step-by-Step Calculation with Clarity", "The total number of possible 6-month funding sequences, given three sectors each month, is:", "> $3^6 = 729$", "Each sequence reflects a unique path the investor might take, from AI to biotech to clean energy.", "To count favorable outcomes—those with exactly two AI, two biotech, two clean energy—we calculate how many sequences yield that precise allocation.", "This is a multinomial coefficient:", "$$\n\ ext{Number of favorable outcomes} = \frac{6!}{2! \cdot 2! \cdot 2!} = \frac{720}{8} = 90\n$$", "These 90 outcomes represent all the ways to"]