Odysseus spends 10 years wandering after the Trojan War. The Odyssey begins after the Greeks have captured Troy, following Odysseus as he struggles for another decade to reach Ithaca. His voyage is repeatedly delayed by storms, divine anger, shipwrecks, and encounters with supernatural beings.
During those years he faces the Cicones, the Lotus-Eaters, the Cyclops Polyphemus, Aeolus, the Laestrygonians, Circe, the Underworld, the Sirens, Scylla and Charybdis, and the island of the Sun. He spends a year with Circe and seven years held by the nymph Calypso before finally leaving for home.
The number can cause a common mix-up because the Trojan War itself also lasts 10 years. Odysseus is therefore absent from Ithaca for roughly 20 years in total: 10 years fighting at Troy and 10 years attempting to return. The quiz’s wording specifically asks about the postwar wandering, so 10 is the intended answer.