3^5 - \binom{3}{1 \cdot 2^5} + \binom{3}{2 \cdot 1^5} = 3^5 - 3 \cdot 2^5 + 3 \cdot 1^5

February 22, 2026

$3^5 - \binom{3}{1 \cdot 2^5} + \binom{3}{2 \cdot 1^5} = 3^5 - 3 \cdot 2^5 + 3 \cdot 1^5$

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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:

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