You have five cards numbered 1,...,5. You shuffle them, so that at the end every permutation of them is equally likely to show up. What is the probability that no card 'i' will end up in the i-th card after shuffling?
So I'm figuring out this problem. I got stuck because the probability of relocating card A changes the probability of moving other cards. For example, if card 1 is moved to location 2 after shuffling, moving card 2 to another location is 4/5, while if card 1 is moved to location 3, moving card 2 to another location will have a probability of 3/5. I'm confused here. How do I solve this problem? Thank you.
Look up "problem of derangements".