Ten people, all wearing hats, were walking along a street when a sudden wind blew their hats off. A helpful boy retrieved them and, without asking which hat belonged to which person, handed each person a hat. What is the probability that exactly 9 of the people received their own hats?

This is what is known as a derangement.

Think about it. If exactly 9 of the 10 get their own hats, then the 10th must also get his own hat. It's the only one left.

Its derivation is rather complicated, but the probability that exactly k of N men get their hats back is \frac{e^{-1}}{k!}, \;\ as \;\ N\to {\infty}

The probability none get their hat back is works out to be 1/e.

THEREFORE, THE PROB. THAT EXACTLY 9 GE THEIR WON HATS IS 0. Because if 9 get theirs, 10 must get theirs.