Let X = \ 1, 2, 3, 4, 5, 6, 7, 8\ , Y = \ 1, 2, 3, 4\ and Z = \ 3,...

The total number of subsets of $X$ is $2^{n(X)} = 2^8 = 256$.

We need to find the number of subsets $C$ that have a non-empty intersection with both $Y$ and $Z$. We can use the Principle of Inclusion-Exclusion to find the number of subsets that fail this condition.

A subset $C$ fails the condition if $C \cap Y = \emptyset$ or $C \cap Z = \emptyset$.

Case 1: $C \cap Y = \emptyset$
This means $C$ must be a subset of $X \setminus Y = \{5, 6, 7, 8\}$.
The number of such subsets is $2^4 = 16$.

Case 2: $C \cap Z = \emptyset$
This means $C$ must be a subset of $X \setminus Z = \{1, 2, 7, 8\}$.
The number of such subsets is $2^4 = 16$.

Case 3: $C \cap Y = \emptyset$ and $C \cap Z = \emptyset$
This means $C$ must be a subset of $X \setminus (Y \cup Z) = \{7, 8\}$.
The number of such subsets is $2^2 = 4$.

By the Principle of Inclusion-Exclusion, the number of subsets that fail the condition is:
$n(C \cap Y = \emptyset \text{ or } C \cap Z = \emptyset) = 16 + 16 - 4 = 28$.

Therefore, the number of subsets $C$ that satisfy both conditions is:
$256 - 28 = 228$.

Answer: $228$