Solution: Use inclusion-exclusion. Total assignments: $3^6 = 729$. Subtract assignments missing at least one category: $3 \times 2^6 = 192$. Add back assignments missing two categories: $3 \times 1^6 = 3$. Valid assignments: $729 - 192 + 3 = 540$. \boxed{540}Question: An electrical engineer tests a new circuit with 5 identical resistors, each with a 10% probability of failing independently. What is the probability that exactly 2 resistors fail?

["Understanding Probability with Inclusion-Exclusion: Inspired by Electrical Engineering Failures", "In engineering, particularly in electrical design, understanding system reliability is crucial. When dealing with redundant components—like resistors in a circuit—probability models help predict performance under failure conditions. While one common analytical method is the inclusion-exclusion principle, today we explore a practical application inspired by such probabilistic reasoning: calculating the likelihood of component failures.", "---", "### Problem: Resistors in a Circuit", "An electrical engineer tests a circuit using 5 identical resistors. Each resistor independently has a 10% chance of failing. We’re asked: What is the probability that exactly 2 resistors fail?", "This question mirrors inclusion-exclusion in combinatorics—where we account for overlapping failure scenarios—but here, it’s framed through real-world risk assessment.", "---", "### Step 1: Model the Scenario", "- Total resistors: $ n = 5 $\n- Each resistor: probability of failure $ p = 0.1 $, success $ q = 1 - p = 0.9 $\n- Relevant event: exactly 2 out of 5 fail", "This is a classic binomial probability situation:\n$$\nP(\ ext{exactly } k \ ext{ failures}) = \binom{n}{k} p^k (1-p)^{n-k}\n$$", "Applying values:\n$$\nP(\ ext{exactly 2 failures}) = \binom{5}{2} (0.1)^2 (0.9)^3\n$$", "Calculate step-by-step:", "- $ \binom{5}{2} = 10 $\n- $ (0.1)^2 = 0.01 $\n- $ (0.9)^3 = 0.729 $", "So:\n$$\nP = 10 \ imes 0.01 \ imes 0.729 = 0.0729\n$$", "---", "### Enhancing with Combinatorics: The Inclusion-Exclusion Perspective", "Though the binomial formula directly gives the result, we can interpret the failure event through the lens of inclusion-exclusion to reinforce mathematical reasoning—especially useful when modeling complex systems with overlapping conditions.", "Imagine events $ A_1, A_2, \dots, A_5 $, where each $ A_i $ is “resistor $ i $ fails.” We want the probability that exactly 2 resistors fail.", "Using inclusion-exclusion to compute valid failure patterns:", "1. Subtract all assignments where at least one resistor fails:\n $$\n \binom{5}{1} \cdot 0.1 \cdot (0.9)^4 = 5 \ imes 0.1 \ imes 0.6561 = 0.32805\n $$\n But this overcounts cases with 2 failures.", "2. Add back intersections where 2 resistors fail (double-counted in step 1):\n $$\n \binom{5}{2} \cdot (0.1)^2 \cdot (0.9)^3 = 10 \cdot 0.01 \cdot 0.729 = 0.0729\n $$", "3. Higher intersections (three or more failures) are negligible here, since 3 failures exceed the "exactly 2" criterion.", "So net probability:\n$$\nP(\ ext{exactly 2 fail}) = 0.0729\n$$", "This confirms the earlier binomial computation.", "---", "### Final Result", "The probability that exactly 2 out of 5 resistors fail, each with a 10% failure rate, is:\n$$\n\boxed{0.0729}\n\quad \ ext{or } \boxed{7.29%}\n$$", "---", "### Why This Matters", "Just as inclusion-exclusion corrects overcounts in combinatorics, probabilistic models like this underpin reliability engineering. Engineers use such calculations to design fault-tolerant circuits, estimate maintenance needs, and improve system safety.", "When even small resistors fail with 10% chance, understanding exactly k failing waves the empirical alarms—critical in precision electronics.", "---", "Summary:\n- Total setups: $ 3^5 = 729 $ (general inclusion-exclusion case)\n- Valid "exactly 2 failures" via binomial: $ \binom{5}{2}(0.1)^2(0.9)^3 = 0.0729 $\n- Insight: Complex systems require layered probability reasoning—echoing inclusion-exclusion philosophy.", "---", "Perfect for engineers modeling redundancy, failure modes, and system robustness—because math doesn’t just solve numbers, it guides design."]