I was wondering if you could possibly give me a shove in the right direction in solving this combinatorics problem. My main issue is that I struggle a bit with recognising what I am meant to do in longer, more text heavy questions. In this particular instance, I am struggling to see what theorem/formula/idea to proceed with trying. I would really appreciate a small nudge to the effect of "X tells you Y, what can you do with that?" with some cursory explaination of how you got to that. My end goal is to improve with this style of problem and, maybe, seeing how you guys think might prove useful.
I appreciate if this is not the normal style of request for this site and thank you in advance for those who do take the time to respond. Also, sorry for the poor quality title, I was rather unsure what to title this post. If anyone has a better idea, feel free to edit!
Question text
"A university department has $100$ professors. Water damage destroys all but
$90$ offices. The university now has to distribute keys to the professors such that no
matter which $90$ professors come to the university, they can assign the offices to
themselves such that everyone can open their office. In addition, it is strictly forbidden to share
or trade keys. Of course, each professor can get multiple keys and keys to an office can
be given to more than one professor. Show that the minimum number of keys to distribute to the professors is $990$"
Best Answer
Well, with your clarification, we shall have to go by the scenario that out of the $90$ lucky persons whose ofiice is intact, if even one is missing , the "unlucky" person replacing her just has her own useless key (which we can as well consider destroyed), and since any of the lucky $90$ may be the missing person, has to hold the keys to each of the $90$ offices.
Thus minimum number of keys needed $=10*90 +90 =990$