1. We need the number of ordered pairs $(Y,Z)$ such that

   • $Y \subseteq X$
   • $Z \subseteq X$
   • $Y \cap Z = \varnothing$

   where $X=\{1,2,3,4,5\}$.

2. Since $Y$ and $Z$ are disjoint, each element of $X$ has exactly 3 choices:

   • it belongs to $Y$
   • it belongs to $Z$
   • it belongs to neither $Y$ nor $Z$

   It cannot belong to both because $Y \cap Z = \varnothing$.

3. Since $X$ has $5$ elements, total number of ordered pairs $(Y,Z)$ is $3^5 = 243.$

4. Because the pair is ordered, $(Y,Z)$ and $(Z,Y)$ are counted separately when $Y \neq Z$.

5. Therefore, the required number is $243.$

6. Now compare with the given options:

   • A: $35$
   • B: $25$
   • C: $53$
   • D: $52$

   None of these matches $243$.

So the mathematically correct answer is $243$, which is not among the listed options. Hence the stored answer appears to be incorrect.