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:
