+295 Human beings have discovered an inhabitable planet and soon after, they find 10 more inhabitable planets. Of these 11, only 5 are deemed ``Earth-like'' in its resources and the rest are deemed ``Mars-like'' since they lack many important resources. Assume that planets like Earth take up 2 units of colonization, while those like Mars take up only 1. If humanity mobilizes 12 total units of colonies, how many different combinations of planets can be occupied if the planets are all different from each other?
+6248 ok, this one is a bit involved. 6 Mars like planets that use 1 colony unit, 5 Earth like planets that use 2 colony units
12 total colony units.
First off we note that we only consider combinations with even numbers of Marslike planets. Otherwise we would have a colonly unit left over. We also note that we must colonize at least 2 Martian like planets and can only colonize up to the 6 existing Marslike planets. Below is a table summarizing the results.
| Marslike | Earthlike | #combos |
|---|---|---|
| 2 | 5 | 6C2 * 5C5 = 15 |
| 4 | 4 | 6C4 * 5C4 = 75 |
| 6 | 3 | 6C6 * 5C3 = 10 |
Totalling the right column we see there are 100 different combinations using all 12 of the colony units.
