The number of smells is extremely large and probably incalculable. There is a finite amount of matter in the universe, thus a finite amount of olfactory sensations.
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There are a finite number of apartments. Finite numbers may be large or small. There are a finite number of states. The number of molds in my fridge is not exactly finite.
The number of stars is finite.
It is a set which contains a finite number of elements.
I would guess that is because it has a finite number of different states. (It is also known as a finite-state machine.)