This is equivalent to counting the number of unordered partitions of a 4-element set into up to 2 indistinct subsets.
Related Articles
- Therefore, the number of such sequences is $\boxed{20480}$.
- Question: A home-schooled student models planetary alignments using 4 distinguishable satellites and 2 identical orbital slots. Each slot can hold any number of satellites, but the slots are indistinguishable. How many distinct configurations are possible?
- Solution: We are distributing $4$ distinguishable satellites into $2$ indistinguishable orbital slots, where each slot can hold any number (including zero), and the order within a slot does not matter (since slots are indistinct and not ordered).
- Since the slots are indistinct, we count partitions of the set of 4 satellites into at most 2 non-empty indistinct subsets. We consider all partitions of 4 labeled objects into 1 or 2 non-empty indistinct subsets.
- Partitions into 1 subset: only $1$ way — all satellites in one slot.
- Partitions into 2 non-empty indistinct subsets: this is the number of equivalence classes under swapping, given by the Stirling number of the second kind $S(4,2)$, which counts the number of ways to partition 4 distinguishable objects into 2 non-empty indistinct subsets.
