Problem: An engineering firm requires all employees to create a secure 4-character PIN. The first character must be a non-zero digit (1-9), and the remaining three characters can be any digit (0-9). How many distinct PINs can be created?

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Problem: Five civil engineering students (Alice, Bob, Charlie, Diana, and Eve) are lining up to present their senior design projects. In how many different orders can they present if Alice insists on presenting first?

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Problem: A city planning department has 12 engineers. They need to form a specialized task force consisting of 5 engineers to evaluate a new bridge design. How many different task forces can be formed?

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Problem: A structural engineer is evaluating two independent sensor systems on a dam. System A has a $95\%$ probability of functioning correctly during a seismic event ($P(A) = 0.95$). System B has a $90\%$ probability of functioning correctly ($P(B) = 0.90$). What is the probability that both systems fail during a seismic event?

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Problem: A batch of 100 concrete cylinders is tested. 10 cylinders fail the compressive strength test. Of those 10 failures, 7 were also found to have been improperly cured. If a cylinder is selected at random and is known to have failed the compressive test, what is the probability that it was improperly cured?

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