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📅 May 5, 2026 👤 scraface

May 05, 2026

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📌 We are to count the number of binary sequences of length 8 with exactly 3 P's (positive) and 5 N's (negative), such that no two P's are adjacent.

📌 First, place the 5 N's. This creates 6 possible "gaps" where P's can be placed — one before the first N, one after each N (between N's), and one after the last N. For example:

📌 \(_ \, N \, _ \, N \, _ \, N \, _ \, N \, _ \, N \, _ \) → 6 gaps.

📌 We must place 3 P's into these 6 gaps, with at most one P per gap (to prevent adjacency).

📌 The number of ways to choose 3 distinct gaps from 6 is:

📌 \binom{6}{3} = 20

📌 Each choice corresponds to a unique sequence where no two P’s are adjacent and exactly 3 are positive.

📌 Thus, the number of valid test result sequences is \(\boxed{20}\).

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