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📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 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).

📌 This is equivalent to counting the number of unordered partitions of a 4-element set into up to 2 indistinct subsets.

📌 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.

📌 We compute $S(4,2)$:

📌 $S(4,2) = 2^2 - 1 = 7$? No — standard recurrence:

📌 $S(3,2) = 2 \cdot 1 + 1 = 3$

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