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An angel investor is evaluating 8 early-stage biotech startups. In how many ways can she choose 3 startups to receive initial funding if the order of selection does not matter?
To solve this, we need to calculate the number of combinations of 8 startups taken 3 at a time. This is given by the combination formula:
Substituting \( n = 8 \) and \( r = 3 \):
\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56
Thus, there are 56 ways to choose 3 startups from 8.
\boxed{56}
An urban forestry researcher is analyzing tree species in a park with 4 distinct types of trees. If each tree can be either healthy or unhealthy, how many different health status combinations are possible across a row of 5 trees?
Each tree has 2 possible states: healthy or unhealthy. Since there are 5 trees and each tree's status is independent, the total number of combinations is: