TW-TST 1999 Problem 6

There are eight different symbols designed on $n\geq 2$ different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any $k$ symbols $(1\leq k\leq 7)$ the number of shirts containing at least one of the $k$ symbols is even. Determine the value of $n$ .