I think you’re right; probably I overcounted one of the 12’s as a 24. (Of course this was so long ago I don’t remember where it went wrong!)

I’ll make little pictures, so people can more easily see what we’re saying. Here are the nine possibilities you discuss, in the same order. We’re just showing the tethers.

| | \/ | | | | \/ | | | \/ | \|/ | | | | | | | | | | | | | | | | | ================================= 24 12 24 24 12 12 12 4 1

In each case the number is 24 — the number of ways to assign four objects into four roles — divided by some symmetries in the arrangement — the ways to interchange various roles. For example in the second-to-last arrangement, there are six ways to shuffle around the three tethers out the top.

(Notation: astronauts A, B, C, and D; ship S; A > B means A is tethered to B)

A > B > C > D > S (24 variations)

B > C > D > S; A > C (12 variations)

B > C > D > S; A > D (24 variations)

B > C > D > S; A > S (24 variations)

C > D > S; A > B > S (12 variations)

C > D > S; A > D; B > S (12 variations)

C > D > S; A > S; B > S (12 variations)

A > S; B > A; C > A; D > A (4 variations)

A > s; B > S; C > S; D > S (1 variation)

This adds up to a total of 125 variations; am I overlooking something? Thanks!

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