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Home » The number of ways is $\boxed243$. - Abbey Badges

The number of ways is $\boxed243$. - Abbey Badges

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 S(n,2) = 2^{n-1} - 1

📌 For \(n = 7\):

📌 S(7,2) = 2^{6} - 1 = 64 - 1 = 63

📌 Alternatively, each object can go into one of the two subsets, giving \(2^7 = 128\) labeled assignments, but dividing by 2 for indistinguishability gives \(64\), and subtracting the two cases where all are in one zone (leaving the other empty), we get \(64 - 1 = 63\) (we subtract 1 for the all-empty-one-case, but actually, each symmetric split is counted once, so the standard result applies directly).

📌 Thus, the number of distinct groupings is \(\boxed{63}\).

📌 Question:** A data analyst in Toronto is analyzing sequences of 8 diagnostic test results, each being either positive (P) or negative (N). How many such sequences contain exactly 3 positive results with no two consecutive positives?

📌 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:

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