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 $n$.
1. **Define the sets and mappings:**
   - Let \( S \) be the set of symbols with \( |S| = 8 \).
   - Let \( \mathcal{T} \) be the set of T-shirts with \( |\mathcal{T}| = t \).
   - Define an injection \( i: \mathcal{T} \to \mathcal{P}(S) \setminus \{\emptyset\} \) by \( i(T) = \{s \in S \mid s \in T\} \).
   - Define \( j: S \to \mathcal{P}(\mathcal{T}) \) by \( j(s) = \{T \in \mathcal{T} \mid s \in T\} \).

2. **Given conditions:**
   - For any set \( \emptyset \neq K \subsetneq S \), the number of T-shirts containing at least one symbol from \( K \) is even.
   - This implies \( |j(s)| \) is even for any \( s \in S \).

3. **Induction on \( |K| \):**
   - By induction on \( |K| \), we show \( \left| \bigcap_{s \in K} j(s) \right| \) is even for all \( 1 \leq |K| \leq 7 \).
   - Using the Principle of Inclusion-Exclusion (PIE):
     \[
     \left| \bigcup_{s \in K} j(s) \right| = \sum_{\ell=1}^{|K|} (-1)^{\ell-1} \sum_{\substack{|L| = \ell \\ L \subseteq K}} \left| \bigcap_{s \in L} j(s) \right|
     \]
     Since \( \left| \bigcup_{s \in K} j(s) \right| \) is even, the right-hand side must also be even.

4. **Maximal set \( M \):**
   - Consider \( M \in \mathcal{T} \) with maximal \( |i(M)| > 0 \).
   - If \( |i(M)| < 8 \), then \( M \in \bigcap_{s \in i(M)} j(s) \).
   - Since \( \left| \bigcap_{s \in i(M)} j(s) \right| \) is even, there exists another \( T \neq M \) with \( i(M) \subsetneq i(T) \), contradicting the maximality of \( |i(M)| \).
   - Therefore, \( |i(M)| = 8 \), i.e., \( i(M) = S \).

5. **Induction on \( k \):**
   - Assume for some \( 1 \leq k < 8 \), \( i^{-1}(S') \in \mathcal{T} \) for all \( S' \subseteq S \) with \( |S'| > k \).
   - For \( K \subset S \) with \( |K| = k \), \( \bigcap_{s \in K} j(s) \) contains \( 2^{8-k} - 1 \) T-shirts.
   - For parity reasons, it must contain at least one more element, which can only be \( i^{-1}(K) \).
   - Thus, \( i^{-1}(K) \in \mathcal{T} \) for all \( K \subset S \) with \( |K| = k \).

6. **Conclusion:**
   - By induction, \( i \) is surjective, so \( t = |\mathcal{T}| = |\mathcal{P}(S) \setminus \{\emptyset\}| = 2^8 - 1 = 255 \).

The final answer is \( \boxed{ n = 8 } \).