**The Secret to Aroc Stocks Explosive Returns โ€” Why No One

AROC Stock Price and Chart โ€” NYSE:AROC โ€” TradingView

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AROC Stock Price and Chart โ€” NYSE:AROC โ€” TradingView
AROC Stock Price and Chart โ€” NYSE:AROC โ€” TradingView
AROC Stock Price and Chart โ€” NYSE:AROC โ€” TradingView
AROC Stock Price and Chart โ€” NYSE:AROC โ€” TradingView
AROC Stock Price and Chart โ€” NYSE:AROC โ€” TradingView

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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 $\boxed{36}$.

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