Each person attending a party has been asked to bring a prize. The person planning the party has arranged to give out exactly as many prizes as there are guests, but any person may win any number of the prizes. If there are $n$ guests, in how many ways may the prizes be given out so that nobody gets the prize that they brought?
My answer was $(n-1)^n$, for I thought that there are $(n-1)$ possible prizes for each of the $n$ guests. However, my answer is wrong. I'm given a hint that it will involve with the inclusion-exclusion principle, I'm not sure how to interpret the problem in this direction.