Solution: The total number of ways to choose 3 items is $\binom{16}{3} = 560$. The favorable cases are $\binom{7}{2} \times \binom{9}{1} = 21 \times 9 = 189$. The probability is $\frac{189}{560} = \frac{27}{80}$. Thus, the answer is $\boxed{\dfrac{27}{80}}$.

["Understanding Combinatorics: How Combinations Solve Probability Problems", "When solving probability problems involving selections from large sets, combinatorics provides a powerful mathematical foundation. One classic example involves calculating the probability of selecting specific items from a group—whether in games, statistics, or everyday decision-making.", "The Basic Formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$", "Combinations count the number of ways to choose $k$ items from $n$ without regard to order. For instance, determining how many ways you can choose 3 items from a total of 16 is calculated using the binomial coefficient:\n$$\n\binom{16}{3} = \frac{16!}{3!(16-3)!} = \frac{16 \ imes 15 \ imes 14}{3 \ imes 2 \ imes 1} = 560\n$$\nThis tells us there are 560 possible groups of 3 selected from 16 items.", "Favorable Outcomes: Breaking the Problem Down", "In probabilité questions, we often focus on favorable outcomes—the specific ways desired conditions are met. Suppose in this problem, 7 items meet a certain criterion and we wish to pick exactly 2 from this group, while the third item comes from the remaining 9. This splits the favorable outcomes into two independent choices:", "- Choose 2 items from the 7: $\binom{7}{2} = 21$\n- Choose 1 item from the other 9: $\binom{9}{1} = 9$", "Multiplying these gives the total favorable combinations:\n$$\n\binom{7}{2} \ imes \binom{9}{1} = 21 \ imes 9 = 189\n$$", "Calculating the Probability", "The probability of the favorable outcomes is simply the ratio of favorable cases to total possible cases:\n$$\n\frac{189}{560}\n$$\nSimplifying this fraction: divide numerator and denominator by 7:\n$$\n\frac{27}{80}\n$$\nSo the final probability is:\n$$\n\boxed{\dfrac{27}{80}}\n$$", "Why This Matters", "Understanding this process helps in fields ranging from statistics and computer science to game theory. Whether analyzing randomized systems or making strategic choices, combinatorial reasoning ensures accurate and efficient calculation of multi-stage selections.", "Key Takeaways:\n- Use $\binom{n}{k}$ to count combinations.\n- Break complex selection problems into manageable parts.\n- Simplify fractions to present clear probability answers.", "This elegant application of combinatorics proves how mathematical principles underpin real-world probability reasoning."]