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A biotech incubator hosts 6 startups. Each startup can either present to a panel of investors on Monday or Tuesday. How many different schedules are possible if exactly 4 startups present on Monday and the remaining 2 on Tuesday?
We need to count the number of ways to choose 4 startups out of 6 to present on Monday (the other 2 automatically go to Tuesday). This is a combination:
\binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6 \times 5}{2 \times 1} = 15
Thus, there are 15 possible schedules.
\boxed{15}
An urban planner is designing a walking path through a city park with 7 distinct native plant zones. If the path must include exactly 4 zones and no zone is repeated, how many different paths (sequences) of zones are possible?
Since the order of visiting the zones matters (it’s a path), we compute the number of permutations of 7 zones taken 4 at a time:
P(7, 4) = \frac{7!}{(7-4)!} = \frac{7!}{3!} = 7 \times 6 \times 5 \times 4 = 840
So, there are 840 distinct possible walking paths.