Problem:
Eight people are attending a seminar in a room with eight chairs. In the middle of the seminar, there is a break and everyone leaves the room.
a) In how many ways can the group sit down after the break so that no-one is in the same chair as before?
b) In how many ways can the group sit down after the break so that exactly four people are in the same chair as before?
c) In how many way can the group sit down after the break so that at least four people are in the same chair as before?
Note that your answer in each case should be an integer.
For any answer you give, describe your solution process in words.
For proofs, write out the proof as you normally would. Then, include an explanation of your proof PRIMARILY IN WORDS. So you should have one regular proof (the kind you see in textbooks), then another proof that describes the proof mainly in words(you don't see this that often).
Provide complete and step by step solution for the question and show calculations and use formulas.