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Home » Let Width = \( x \), Length = \( 2x \) - 500apps

Let Width = \( x \), Length = \( 2x \) - 500apps

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Number of ways: \(\binom{6}{3} = 20\)

📌 Each such choice gives a unique valid configuration.

📌 So the number of valid configurations is \(\boxed{20}\)

📌 A virologist is analyzing sequences of 7 amino acids, each being either hydrophobic (H) or hydrophilic (P). How many sequences contain exactly four H’s, with no two H’s adjacent?

📌 We need to place 4 H’s and 3 P’s in 7 positions, such that no two H’s are adjacent.

📌 First, place the 3 P’s. This creates 4 possible gaps (before, between, after) where H’s can go:

📌 \_ P \_ P \_ P \_ → 4 gaps

📌 We need to place 4 H’s into these 4 gaps, with **at most one H per gap** to prevent adjacency. But we have 4 H’s and only 4 gaps, so we must place exactly one H in each gap.

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