Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the mininum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that
\[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\]
Evaluate $M(200) \cdot m(200)$.
1. **Understanding the problem**: We need to find the maximum and minimum number of planes required to divide $\mathbb{R}^3$ such that no region contains more than one point from a set $S$ of 200 points. We denote these values as $M(200)$ and $m(200)$ respectively, and we need to evaluate $M(200) \cdot m(200)$.

2. **Maximum number of planes, $M(n)$**: 
   - To ensure that no two points from $S$ are in the same region, we need to create at least $n$ regions.
   - The maximum number of regions $R(k)$ that $k$ planes can divide $\mathbb{R}^3$ into is given by:
     \[
     R(k) = \binom{k}{3} + \binom{k}{2} + \binom{k}{1} + \binom{k}{0}
     \]
   - For $n = 200$, we need $R(k) \geq 200$. We need to find the smallest $k$ such that this inequality holds.
   - Calculating $R(k)$ for various values of $k$:
     \[
     \begin{aligned}
     R(1) &= 1 + 1 = 2, \\
     R(2) &= 1 + 2 + 1 = 4, \\
     R(3) &= 1 + 3 + 3 + 1 = 8, \\
     R(4) &= 1 + 4 + 6 + 4 + 1 = 15, \\
     R(5) &= 1 + 5 + 10 + 10 + 5 + 1 = 26, \\
     R(6) &= 1 + 6 + 15 + 20 + 15 + 6 + 1 = 42, \\
     R(7) &= 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 = 64, \\
     R(8) &= 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 93, \\
     R(9) &= 1 + 9 + 36 + 84 + 126 + 126 + 84 + 36 + 9 + 1 = 130, \\
     R(10) &= 1 + 10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 + 1 = 176, \\
     R(11) &= 1 + 11 + 55 + 165 + 330 + 462 + 462 + 330 + 165 + 55 + 11 + 1 = 232, \\
     \end{aligned}
     \]
   - We see that $R(11) = 232 \geq 200$, so $M(200) = 11$.

3. **Minimum number of planes, $m(n)$**:
   - The minimum number of planes required to ensure that no two points from $S$ are in the same region is $n-1$.
   - For $n = 200$, we have $m(200) = 200 - 1 = 199$.

4. **Calculating the product**:
   \[
   M(200) \cdot m(200) = 11 \cdot 199 = 2189
   \]

The final answer is $\boxed{2189}$.