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

February 22, 2026

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

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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.
Question:** A climate therapist in Boston is designing 5 distinct mindfulness sessions using 3 sensory environments: forest sounds, ocean waves, and forest silence. Each session uses exactly one environment, and each environment must be used at least once. How many distinct schedules are possible?
We are assigning 5 distinguishable sessions to 3 distinguishable environments (forest, ocean, silence), such that each environment is used at least once. Since the sessions are distinct, this is a **surjective function** from a 5-element set to a 3-element set, with labeled outputs.
The number of such surjective functions is given by:

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