Solution: Total items: $ 4 + 6 + 2 = 12 $. Total ways to choose 2: $ \binom{12}{2} = 66 $. Favorable ways: $ \binom{6}{2} = 15 $. Probability: $ \frac{15}{66} = \frac{5}{22} $.

["Probability Explained: Choosing 2 Items from a Set in 66 Ways", "When selecting items from a collection, understanding the probability of specific outcomes helps make informed decisions—whether you're planning strategies, analyzing data, or playing games. This article breaks down a classic probability scenario involving combinations, walks through the key calculations, and explains how to arrive at a clear and accurate probability.", "---", "### The Scenario: Picking 2 Items from a Diverse Set", "Suppose you have a set of 12 distinct items composed of different subsets:\n- 4 blue items\n- 6 green items\n- 2 red items", "You want to know:\n- How many total ways are there to choose any 2 items from this set?\n- How many favorable outcomes exist where both selected items are green?\n- Finally, what is the probability of picking two green items?", "---", "### Step 1: Total Number of Ways to Choose 2 Items", "The total number of ways to choose 2 items from 12 is given by the combination formula:\n[\n\binom{12}{2} = \frac{12!}{2!(12 - 2)!} = \frac{12 \ imes 11}{2} = 66\n]\nSo, there are 66 total possible combinations when selecting any 2 items from 12.", "---", "### Step 2: Number of Favorable Ways (Choosing 2 Green Items)", "Only 6 items are green, and you’re interested in choosing 2 of them. The number of ways to choose 2 green items is:\n[\n\binom{6}{2} = \frac{6!}{2!(6 - 2)!} = \frac{6 \ imes 5}{2} = 15\n]\nThus, there are 15 favorable combinations where both chosen items are green.", "---", "### Step 3: Calculating the Probability", "Probability is calculated as:\n[\n\ ext{Probability} = \frac{\ ext{Favorable ways}}{\ ext{Total ways}} = \frac{\binom{6}{2}}{\binom{12}{2}} = \frac{15}{66}\n]\nThis fraction simplifies to:\n[\n\frac{15 \div 3}{66 \div 3} = \frac{5}{22}\n]", "---", "### Conclusion: The Probability is ( \frac{5}{22} )", "There are 66 total ways to choose any 2 items from 12, and 15 favorable ways where both items are green. The probability of selecting two green items is therefore:\n[\n\boxed{\frac{5}{22}}\n]", "Understanding combinations and probability not only helps solve math problems but also empowers data-driven decisions in real-world applications—from sports analytics to quality control in manufacturing. By breaking down the problem step by step, we ensure clarity and accuracy in probability calculations."]