**An astronomer observes an exoplanet that orbits its star every 42 Earth days. If a second exoplanet in the same
Solution: The total number of ways to choose 3 studies from 8 is:
To count the favorable outcomes where the impactful study is included, we fix that study and choose 2 more from the remaining 7:
Thus, the probability that the impactful study is selected is:
Question: An electrical engineer designing a high-voltage transformer notes that two circuits reset every 18 and 24 seconds. After how many seconds will they both reset simultaneously?
Question: A hydrologist models groundwater flow and finds that two aquifers release water every 30 and 45 days. What is the greatest common divisor of these intervals, representing the largest periodic overlap?
Question: A primatologist records primate calls in base-6. What is the base-ten equivalent of the base-six number $432_6$?
A researcher is tracking a population of a rare bird species across four distinct regions. She captures and tags 5 birds in each region. If she wants to choose 3 tagged birds for a detailed study, in how many ways can she select them such that at least one bird comes from each region?
The total number of ways to choose any 3 birds from 20 is given by
We subtract the cases where the birds do not come from all regions. This involves:
All 3 birds from the same region (impossible since there are only 5 in each region, and we need 3 from one).