Not fake. Two ways to collect matchbooks - with or without matches. Neither method being "right" or "wrong" but a seller should be specific on what he's offering! WITH matches - fire hazard and perhaps against postal regulations, and not a great thing to tell your insurer that you collect. WITHOUT - easier to handle and ship. BUT, if the matches themselves have some design printed on them, they should be left in the books.

I once got in some matchbooks to sell on consignment, and refused to list any of the lot, for the concerns voiced by others answering here.