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mystery case files - Decision Point

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Solve for w: \(w = 64 / 8 = 8\).

📌 Length = 3w = 3 * 8 = 24.

📌 #### Width = 8 m, Length = 24 mQuestion: How many distinct ways can a STEM advocate schedule 5 different workshop sessions if two specific sessions must be held on consecutive days?

📌 Solution: Treat the two consecutive sessions as a single entity. There are $4!$ ways to arrange the 4 entities (the pair and the other 3 sessions). The pair can be ordered in $2!$ ways. Thus, total arrangements are $4! \times 2! = 24 \times 2 = 48$. \boxed{48}

📌 Question: An angel investor plans to fund 4 startups from a pool of 9, but two specific startups cannot both be funded. How many valid funding combinations are there?

📌 Solution: Total combinations without restrictions: $\dbinom{9}{4} = 126$. Subtract combinations where both problematic startups are included: $\dbinom{7}{2} = 21$ (choosing 2 more from the remaining 7). Valid combinations: $126 - 21 = 105$. \boxed{105}

📌 Question: An epidemiologist models disease spread by assigning 6 unique patient IDs to 3 distinct risk categories, ensuring each category has at least one ID. How many assignments are possible?

📌 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?

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