A Thanksgiving Puzzle
Sixteen people at our Thanksgiving table this year; my sister is hosting again. If all four kids got up at once, and then ran back to the table and sat down in any empty seat, how likely is it that none of them will be sitting in their original seats?
There are two notes, 1) where I adapted this from and 2) some hints about the topic, if you click —————>
Note 1: Adapted from a much more colorful problem offered me last year by Paul Schwegerling, a retired Buffalo math teacher and the director of an enrichment center: Ten sailors hit port after months at sea, and (do whatever sailors do). They stumble back on board days later, falling down drunk. Ten empty bunks, ten drunk sailors, they each fall into one. What is the chance that none of them is in the right bunk?
Note 2: In combinatorics we have covered these as a class of question, reducing them from puzzle to hard exercise. I think, though, that most people never encounter them. At first contact they function as puzzles.
The generalization is interesting. I will write more in the comments section, or as a separate post, later.
JD2718 
I haven’t come up with a neat formula, but it looks to me like there are 24 different ways the kids could re-seat themselves, and in 15 of them at least one kid would be sitting in his/her original place, so the chances that none of them are in the same seat is 9/24.
I’m trying to generalize, but it’s tough. If you imagine the kids or sailors returning one at a time, then the first person has (n-1) acceptable choices. Anywhere except their original spot is fine. But the next person back might have (n-2) or (n-1) choices, depending if the first person happened to pick the next person’s original spot. So I’m guessing that this whole line of analysis, based on considering the seating choices as independent events, isn’t going to help. But I don’t know. I can’t think of any other way to analyze it either.
It’s clear that there are n! possible orderings of n items. It’s less clear how to figure out exactly how many of those orderings have one or more items in the same spot as a particular chosen initial ordering.
rdt’s answer (looks like he listed the possibilities) is correct.
As far as generalizing, would you like:
1) more time
2) the answer
3) a small hint
4) directions to generalize (leaving considerable work) ?
I’d be happy to have a small hint…
We covered this type of problem while discussing inclusion/exclusion.