3^5 - \binom{3}{1 \cdot 2^5} + \binom{3}{2 \cdot 1^5} = 3^5 - 3 \cdot 2^5 + 3 \cdot 1^5
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- 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:
- This is an application of inclusion-exclusion:
- = 243 - 3 \cdot 32 + 3 \cdot 1 = 243 - 96 + 3 = 150
- Alternatively, using Stirling numbers of the second kind:
