Let $S$ be a set of finite number of points in the plane any 3 of which are not linear and any 4 of which are not concyclic. A coloring of all the points in $S$ to red and white is called [i]discrete coloring[/i] if there exists a circle which encloses all red points and excludes all white points. Determine the number of [i]discrete colorings[/i] for each set $S$.
1. **Understanding the Problem:**
   We are given a set \( S \) of \( n \) points in the plane, where any 3 points are not collinear and any 4 points are not concyclic. We need to determine the number of ways to color these points red and white such that there exists a circle enclosing all red points and excluding all white points. This is called a *discrete coloring*.

2. **Mapping Circles to Points in \(\mathbb{RP}^3\):**
   We use a stereographic projection to map the plane to a sphere. Under this projection, circles in the plane map to circles on the sphere. We then define the image of a circle on the sphere to be the inversion of its center about the sphere. This mapping has the following properties:
   - Generalized circles (including lines) in the plane map to points strictly outside the sphere.
   - The set of all circles that contain a point on their exterior map to a plane in \(\mathbb{RP}^3\).
   - A circle contains a point in its interior if and only if its image lies on one side of this plane.

3. **Mapping Points to Planes:**
   By mapping the \( n \) points in the plane to \( n \) planes in \(\mathbb{RP}^3\), we transform the problem into determining the number of regions into which these \( n \) planes divide the space. Each region corresponds to a unique way of coloring the points such that there exists a circle enclosing all red points and excluding all white points.

4. **Counting the Regions:**
   The number of regions into which \( n \) planes divide \(\mathbb{RP}^3\) is given by the sequence [A000125](https://oeis.org/A000125). This sequence is defined as:
   \[
   \sum_{k=0}^{3} \binom{n}{k}
   \]
   where \(\binom{n}{k}\) is the binomial coefficient representing the number of ways to choose \( k \) planes from \( n \) planes.

5. **Calculating the Number of Regions:**
   The number of regions is:
   \[
   \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \binom{n}{3}
   \]
   Let's break this down:
   - \(\binom{n}{0} = 1\)
   - \(\binom{n}{1} = n\)
   - \(\binom{n}{2} = \frac{n(n-1)}{2}\)
   - \(\binom{n}{3} = \frac{n(n-1)(n-2)}{6}\)

   Therefore, the total number of regions is:
   \[
   1 + n + \frac{n(n-1)}{2} + \frac{n(n-1)(n-2)}{6}
   \]

6. **Conclusion:**
   The number of *discrete colorings* for a set \( S \) of \( n \) points is given by:
   \[
   \boxed{1 + n + \frac{n(n-1)}{2} + \frac{n(n-1)(n-2)}{6}}
   \]