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.
This problem can also be expressed in terms of graph theory. If you connect up 5 points (the 4 astronauts and the spaceship) using the smallest number of edges you have created what in graph theory is called a spanning tree. Cayley’s formula (named for Arthur Cayley :1821-1895) says there are n^(n-2) different spanning trees for n points. So for our 5 points there would be 5^(5-2)=5^3=125 different possibilities (as calculated by permutations above). If spaceman Erp joined in there would be 6 points (A,B,C,D,E and the Spaceship) so there will be 6^4 = 1296. To prove this is quite tricky but google Cayley’s Formula for more info.