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Home » The surface area \(SA\) is calculated by: - Redraw

The surface area \(SA\) is calculated by: - Redraw

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 5Question: A research team at the NIH reviews 120 grant applications, of which 45 focus on cancer research, 50 on neuroscience, and 25 on infectious diseases. If three applications are selected at random without replacement, what is the probability that all three selected apply to different fields?

📌 Solution: We are selecting 3 applications at random without replacement from 120 total, and we want the probability that each application is from a distinct field—one from cancer, one from neuroscience, and one from infectious diseases.

📌 The total number of ways to choose any 3 applications is:

📌 The number of favorable outcomes—one from each distinct field—requires selecting one application from each of the three categories. Since selections are without replacement and order doesnt matter in the combination, we compute:

📌 However, since the order in which the fields are selected doesnt matter, we must consider the number of ways to assign the three different fields to the three draws. There are $3! = 6$ permutations of the three fields (cancer, neuroscience, infectious), but since were forming unordered triples, each combination of one from each field is counted exactly once per unique field assignment. Alternatively, since we are selecting one from each group, and the groups are fixed, the number of favorable unordered triples is:

📌 but we must divide by no overcounting factor since we are selecting specific individuals—each person is distinct, so no division is needed in this product approach. Thus, favorable outcomes are simply the product of individual choices:

📌 The factor of 6 accounts for the 6 possible orders in which one can pick one from each field. However, note that the event all three from different fields does not depend on order, so a better approach is to compute the probability directly as:

📌 Total number of ways to choose 3 applications:

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