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📰 Solution: First, calculate the total number of ways to choose 5 opportunities from 8 without restrictions: $\binom{8}{5} = 56$. Next, subtract the number of invalid portfolios that include both mutually exclusive opportunities. If both are selected, the remaining 3 are chosen from the other 6: $\binom{6}{3} = 20$. Thus, the valid portfolios are $56 - 20 = 36$. The final answer is $\boxed{36}$. 📰 Question: A science educator designs a 6-module interactive curriculum using 3 physics simulations, 2 chemistry experiments, and 1 biology lab. How many distinct arrangements are possible if the biology lab must not be placed after the chemistry experiments? 📰 Solution: First, arrange the 6 modules without restrictions: $\frac{6!}{3!2!1!} = 60$. For the constraint, note the biology lab (B) must not follow both chemistry experiments (C). Total valid arrangements: Calculate total permutations where B is not after both C's. This is equivalent to ensuring B is not in a position after both C's. Using combinatorial cases: B is first, or B is second with at least one C before it, or B is third with at least two C's before it. Alternatively, recognize that the condition excludes only $ \frac{1}{3} $ of all permutations where B is after both C's (since the C's can be ordered in 2 ways). Thus, valid arrangements: $60 - \frac{1}{3} \times 60 = 40$. The final answer is $\boxed{40}$. 📰 Who Wins In The Long Walk 7491953 📰 Life Changing Healing Quotes That Will Heal Your Soul In Seconds 2940589 📰 Smc College 6159304 📰 You Wont Guess The Real Power Behind Pokemon Fire Red Leaf Greenfind Out Now 4694790 📰 Gargamel Gargamel Exposed The Dark Truth Behind This Infamous Villain 5857128 📰 How A Simple Sandcastle Changed One Beach Into A Magical Community Story 7784203 📰 How Many Dimensions Are There 9426502 📰 Futures Trading Charts 8307474 📰 Indiana Mens Basketball Coaches 4487735 📰 The Hidden Trick To Finding And Changing Words Instantlyno Guesswork Allowed 5528810 📰 Msos Etf Breakthrough Inside The Hidden Strategy Behind Its Rapid Rise 6838360 📰 Claisen 2937775 📰 Taqueria Juarez 4666142 📰 Why Every Room Needs Color Blocks Proven Benefits That Sprinkle Joy 1535386 📰 The Hidden Danger Inside Mysterious Snails Scientists Stunned By These Hidden Secrets 4942156