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Home » Subtract 6: $ 5x = 96 $. - RoadRUNNER Motorcycle Touring & Travel Magazine

Subtract 6: $ 5x = 96 $. - RoadRUNNER Motorcycle Touring & Travel Magazine

📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 A computational biologist is studying a population of 3 distinct bacterial strains labeled A, B, and C. If a sample of 5 bacteria is taken such that at least one bacterium from each strain is included, how many different samples can be formed?

📌 To solve this problem, we first calculate the total number of ways to choose 5 bacteria from 3 strains without any restrictions, and then subtract the cases where one or more strains are missing.

📌 Total ways to choose 5 bacteria from 3 strains (A, B, C):

📌 Since there are 3 strains, we can treat this as distributing 5 indistinguishable bacteria into 3 distinguishable strains. This is a classic stars and bars problem:

📌 Subtract the cases where at least one strain is missing:

📌 However, lets verify this by listing the valid compositions where each strain appears at least once:

📌 Thus, the number of different samples with at least one bacterium from each strain is:

📌 A high-performance computing cluster at Oak Ridge National Laboratory runs a simulation job that can execute in 6 distinct modes (labeled M1 to M6). If the system is configured to run 4 jobs consecutively, with no two consecutive jobs in the same mode, how many valid execution sequences are possible?

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