Every Friday ten gentlemen come to the club, and each one gives the doorman his hat. Each hat is just right for its owner, but there are no two identical hats. The gentlemen leave one by one in a random order.
Seeing off the next gentleman, the club’s doorman tries to put the first hat that he grabs on the gentleman’s head. If it fits (not necessarily perfectly), the gentleman leaves with that hat. If it is too small, the doorman tries the next random hat from the remaining ones. If all of the remaining hats turned out to be too small, the doorman says to the poor fellow: “Sir, you do not have a hat today,” and the gentleman goes home with his head uncovered. Find the probability that next Friday the doorman will not have a single hat.