Question: A palynologist has 5 red pollen grains, 3 green pollen grains, and 2 blue pollen grains. If she randomly selects 6 grains from the bag without replacement, what is the probability that exactly 2 red, 2 green, and 2 blue grains are selected?

["Understanding the Probability of Selecting Specific Pollen Grains: A Detailed Breakdown", "In palynology, the study of pollen and spores, understanding the probability of selecting specific grain types from a mixed sample is essential for accurate data analysis and ecological interpretation. A common question arises when selecting a subset of pollen grains at random: what is the probability of drawing a balanced sample of certain colors—say, exactly two red, two green, and two blue grains?", "### The Scenario", "Imagine a pollen sample containing:\n- 5 red pollen grains\n- 3 green pollen grains\n- 2 blue pollen grains\nfor a total of 10 pollen grains.", "A palynologist randomly selects 6 grains without replacement—meaning each grain is drawn once and never returned. The question is:", "> What is the probability that exactly 2 red, 2 green, and 2 blue grains are selected?", "---", "### Step 1: Total Ways to Choose 6 Grains from 10", "The total number of ways to choose any 6 grains from 10 is given by the combination formula:", "$$\n\binom{10}{6} = \frac{10!}{6!(10-6)!} = \frac{10 \ imes 9 \ imes 8 \ imes 7}{4 \ imes 3 \ imes 2 \ imes 1} = 210\n$$", "This represents all possible ways to draw a sample of 6 grains from the full set.", "---", "### Step 2: Favorable Outcomes – Selecting Exactly 2 Red, 2 Green, and 2 Blue", "We need to compute the number of favorable combinations where:\n- 2 out of 5 red grains are chosen,\n- 2 out of 3 green grains are chosen,\n- 2 out of 2 blue grains are selected.", "Each of these selections is computed independently using combinations:", "- Ways to choose 2 red from 5:\n $$\n \binom{5}{2} = \frac{5 \ imes 4}{2 \ imes 1} = 10\n $$", "- Ways to choose 2 green from 3:\n $$\n \binom{3}{2} = \frac{3 \ imes 2}{2 \ imes 1} = 3\n $$", "- Ways to choose 2 blue from 2 (only one choice):\n $$\n \binom{2}{2} = 1\n $$", "Multiply these together to get the total number of favorable outcomes:", "$$\n\binom{5}{2} \ imes \binom{3}{2} \ imes \binom{2}{2} = 10 \ imes 3 \ imes 1 = 30\n$$", "---", "### Step 3: Compute the Probability", "Probability is the ratio of favorable outcomes to total outcomes:", "$$\nP(\ ext{2 red, 2 green, 2 blue}) = \frac{\ ext{Favorable outcomes}}{\ ext{Total combinations}} = \frac{30}{210} = \frac{1}{7}\n$$", "---", "### Final Answer", "The probability that exactly 2 red, 2 green, and 2 blue pollen grains are selected when drawing 6 grains at random without replacement is:", "1/7", "---", "### Why This Matters for Palynologists", "This calculation demonstrates how combinatorics underpins sample analysis in palynology. Accurate probability models help researchers interpret vegetation composition, assess sampling bias, and ensure reliable conclusions from microscopic pollen counts. Understanding such probabilities enables more precise ecological monitoring and climate change research using pollen data.", "---", "### Key Takeaways", "- Probability relies on favorable outcomes divided by total possible outcomes.\n- Combinations are used when order does not matter.\n- Sampling without replacement affects total combinations and must be carefully calculated.\n- A balanced sample of 2-2-2 across three color groups, though seemingly simple, has a non-intuitive likelihood encapsulated by numerator 30 over denominator 210.", "---", "By mastering these principles, palynologists enhance the rigor of their field — turning random collections into statistically meaningful insights."]