Five cards numbered 1 to 5 are shuffled and laid out in a row in positions numbered 1 to 5. What is the probability that no card lands in the position matching its own number?

It is far easier to count the arrangements that DO have at least one match, by inclusion–exclusion: C(5,1)4! − C(5,2)3! + C(5,3)2! − C(5,4)1! + C(5,5)0! = 120 − 60 + 20 − 5 + 1 = 76. Of the 5! = 120 arrangements that leaves 120 − 76 = 44 with no match at all, a probability of 44/120 = 11/30. The value 19/30 = 76/120 is the probability of at least one match.