Next, we compute the number of favorable outcomes: selecting exactly 2 red, 2 green, and 2 blue grains. Since there are 5 red grains, 3 green, and 2 blue, the number of favorable combinations is:
["Next, compute the number of favorable outcomes: selecting exactly 2 red, 2 green, and 2 blue grains.\nWith 5 red, 3 green, and 2 blue grains available, determine how many distinct combinations yield exactly 2 grains of each color.", "In probability and combinatorics, counting favorable outcomes is essential for calculating probabilities and understanding possible events. Today, we focus on computing the number of advantageous outcomes when selecting exactly 2 red, 2 green, and 2 blue grains from a limited supply.", "We are given:\n- 5 red grains available,\n- 3 green grains available,\n- 2 blue grains available.", "We want the number of ways to choose exactly:\n- 2 red grains\n- 2 green grains\n- 2 blue grains", "The total number of favorable combinations is found by multiplying the combinations possible for each color, since each choice is independent:", "[\n\ ext{Favorable outcomes} = \binom{5}{2} \ imes \binom{3}{2} \ imes \binom{2}{2}\n]", "Now compute each term:\n[\n\binom{5}{2} = \frac{5!}{2! \cdot 3!} = \frac{120}{2 \cdot 6} = 10\n]\n[\n\binom{3}{2} = \frac{3!}{2! \cdot 1!} = \frac{6}{2 \cdot 1} = 3\n]\n[\n\binom{2}{2} = 1\n]", "Multiply them together:\n[\n10 \ imes 3 \ imes 1 = 30\n]", "Thus, the number of favorable outcomes—selecting exactly 2 red, 2 green, and 2 blue grains—is 30.", "This calculation underpins probability assessments in sampling without replacement, especially in constrained populations. Whether supporting statistical models, quality testing, or game design, understanding how to compute favorable combinations enables precise decision-making and deeper insight into chance processes.", "Keywords: combinatorics, probability calculation, favorable outcomes, math problem solution, binomial coefficients, counting combinations, red green blue grains, coefficient calculation."]
