The trap here is the tumour name itself. Synovial sarcoma sounds like it should grow from the membrane lining a joint, but that idea is wrong. Pathologically it springs from undifferentiated mesenchymal tissue and carries a signature t(X;18) gene fusion, and it merely resembled synovium under the microscope to the people who first named it. Once you accept that it does not come from synovium, the rest of the statements line up logically. Since its source is not the joint lining, it mostly turns up outside joints, in soft tissue near tendons and fascia, so the extra-articular claim holds. It typically strikes teenagers and young adults below the age of fifty, so the age statement holds. And its favourite locations are the deep tissues of the limbs, classically around the knee and the foot, so that statement holds too. The question wants the single false claim, and the false one is that the tumour begins in the synovial lining.
\[\boxed{\text{Originate from synovial lining}}\]