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?

Thank you to Rom and the guest! I really had trouble with this one...

ANotSmartPerson
Nov 14, 2018

#1**+2 **

So distilling all this we have

5 earthlike planets

6 marslike planets

planets are distinct

earthlike planets use 2 units of colonists

marslike planets use 1 unit of colonists

There are 12 units of colonists to assign.

Probably easiest to work it looking at how many earthlike planets are colonized

There are only 6 marslike planets so there must be at least 3 earthlike planets colonized

There are only 5 earthlike planets so there can be at most 5 colonized

\(k ~Earth \Rightarrow 12-2k~Mars\\ N = \sum \limits_{k=3}^5 ~\left[\dbinom{5}{k}\right]\left[\dbinom{6}{12-2k}\right]\\ N=100\)

Rom
Nov 15, 2018