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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Solution: First, choose 4 distinct soil types from 10: $\dbinom{10}{4}$. Then, select 2 chemical treatments from 3: $\dbinom{3}{2}$. Multiply these results: $\dbinom{10}{4} \times \dbinom{3}{2} = 210 \times 3 = 630$. The total number of combinations is $\boxed{630}$.

📌 Question: A science fiction writer designs a starship with 7 crew roles and 5 alien biome systems. How many ways can 3 crew roles and 2 biome systems be selected for a mission if each system requires a unique role?

📌 Solution: First, choose 3 crew roles from 7: $\dbinom{7}{3}$. Then, select 2 biome systems from 5: $\dbinom{5}{2}$. Since each system needs a unique role, multiply by the number of ways to assign roles to biomes: $\dbinom{7}{3} \times \dbinom{5}{2} \times 3! = 35 \times 10 \times 6 = 2100$. However, if roles are independent of biomes, the answer simplifies to $\dbinom{7}{3} \times \dbinom{5}{2} = 35 \times 10 = 350$. Clarifying the problem's constraints, the most logical interpretation is $\boxed{350}$.

📌 Question: A bioinformatics developer creates a tool to analyze 6 gene sequences and 4 protein markers. How many ways can 2 sequences and 2 markers be selected if one specific marker is mandatory for all selections?

📌 Solution: Since one marker is mandatory, choose 1 additional marker from the remaining 3: $\dbinom{3}{1}$. Then, select 2 sequences from 6: $\dbinom{6}{2}$. Multiply these: $\dbinom{6}{2} \times \dbinom{3}{1} = 15 \times 3 = 45$. The total number of valid combinations is $\boxed{45}$.Question: A tech startup founder has 8 potential investment opportunities and wants to diversify by selecting 5, but two specific opportunities are mutually exclusive. How many valid investment portfolios can be formed?

📌 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 $\boxed{40}$.

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