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If \(\mathbfu, \mathbfv, - Roya Kabuki

📅 May 8, 2026 👤 scraface

May 08, 2026

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📌 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.

📌 An angel investor is forming advisory panels by selecting 2 biotech specialists and 2 data scientists from a group of 5 specialists and 6 scientists. How many distinct panels can be formed?

📌 We compute the number of ways to choose 2 specialists from 5 and 2 scientists from 6, then multiply:

📌 \binom{5}{2} = \frac{5!}{2!3!} = \frac{5 \times 4}{2} = 10

📌 \binom{6}{2} = \frac{6!}{2!4!} = \frac{6 \times 5}{2} = 15

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