Example 2. In space, there are 10 points, 4 of which lie on the same plane, and no other set of 4 points are coplanar; find the number of circular cones (not necessarily right circular cones) with one of the points as the vertex and a circle passing through 3 other points as the base.
Analysis: From the problem, we can infer that among the 10 points in space, no 3 points are collinear. Otherwise, each of the remaining 7 points would be coplanar with the 3 collinear points, which contradicts the problem statement. Now, let's denote the plane where 4 points are coplanar as \( a \). We need to consider two cases.

First, without considering the restriction "4 points are coplanar," the 10 points in space, where no 4 points are coplanar, can determine \( C_{10}^{1} C_{0}^{3} \) cones.
( I ) The 4 points on plane \( \alpha \) are concyclic.
In this case, under the restriction, the number of cones that cannot be determined is \( C_{4}^{1} C_{3}^{3} + (C_{4}^{3} - 1) C_{6}^{1} \). According to the complement principle, the number of cones that can be determined is \( C_{10}^{1} C_{0}^{3} - [C_{4}^{1} C_{3}^{3} + (C_{4}^{3} - 1) C_{6}^{1}] \).
(II) The 4 points on plane \( \alpha \) are not concyclic.
In this case, we only need to exclude the cases where the vertex and the 3 points determining the base circle are all on plane \( \alpha \), which is \( C_{4}^{1} C_{3}^{3} \). According to the complement principle, the number of cones that can be determined is \( C_{10}^{1} C_{8} - C_{4}^{1} C_{3}^{3} \).

Solution: Let the plane where 4 points are coplanar in the 10 points in space be \( \alpha \). If no 4 points are coplanar, the 10 points can determine \( C_{10}^{1} C_{9}^{3} \) cones.
( I ) The 4 points on plane \( \alpha \) are concyclic. In this case, under the restriction, the number of cones that cannot be determined is \( C_{4}^{1} C_{3}^{3} + (C_{4}^{3} - 1) C_{8}^{1} \). Therefore, the number of cones that can be determined is
\[
\begin{array}{l}
C_{10}^{1} C_{9}^{3} - [C_{4}^{1} C_{3}^{3} + (C_{4}^{3} - 1) C_{8}^{1}] \\
= 840 - [4 - 18] = 818.
\end{array}
\]
(II) The 4 points on plane \( \alpha \) are not concyclic. In this case, if the vertex and the 3 points determining the base circle are all on \( \alpha \), the number of cones that cannot be determined is \( C_{4}^{1} C_{3}^{3} \). Therefore, the number of cones that can be determined is \( C_{10}^{1} C_{8} - C_{4}^{1} C_{3}^{3} \)
\[
= 840 - 4 = 836.
\]