Very hot! - Lycée Arago (Perpignan) Colegiul National B.P. Hasdeu (Buzau, Roumanie)

A big Christmas party is organized. Every guest must bring a gift and leave it into a big basket when they arrive. Around midnight, Santa offers a present from the basket randomly to each guest. Ideally, everyone should get a present different from the one they brought (we assume that all the gifts are different). What is the chance that it actually happens?