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Question: A historian is analyzing three randomly selected scientific papers from a collection of 12, where 5 are written by Newton and 7 by Leibniz. What is the probability that exactly two of the selected papers are by Newton?
Solution: The total number of ways to choose 3 papers from 12 is:
\binom{12}{3} = 220
To choose exactly 2 papers by Newton and 1 by Leibniz:
\binom{5}{2} \cdot \binom{7}{1} = 10 \cdot 7 = 70
\frac{70}{220} = \frac{7}{22}
\boxed{\frac{7}{22}}
Question: A geotechnical engineer is testing soil samples from 10 different sites, each labeled from 1 to 10. If 3 sites are selected at random for detailed analysis, what is the probability that site 7 is included?
Solution: The total number of ways to choose 3 sites from 10 is:
\binom{10}{3} = 120