A circle can be drawn through any three points on a given plane, provided they are not collinear.

Problem Statement:
There are 24 points on a plane such that 10 of them are collinear. No 4 points are vertices of a cyclic quadrilateral. Find the maximum number of circles that can be drawn through any three points.

Solution:

Understanding the Constraints:
To find the maximum number of circles that can be drawn through any three points, we need to consider the given constraints:
- There are 24 points on a plane.
- 10 of these points are collinear.
- No 4 points are vertices of a cyclic quadrilateral.

Analysis:
To find the maximum number of circles, we need to maximize the number of unique triplets of points that can form a circle. Let's consider the possible scenarios:

Scenario 1: No collinear points
In this scenario, all 24 points are non-collinear. To form a circle, we need 3 non-collinear points. Hence, the maximum number of circles in this scenario is C(24, 3) = 24! / (3! * (24 - 3)!) = 2024.

Scenario 2: Maximum collinear points
In this scenario, we assume that all 10 collinear points are part of every circle. To form a circle, we need 3 points. We can choose 3 points from the collinear set in C(10, 3) ways. For the remaining 14 non-collinear points, we can choose 3 points in C(14, 3) ways. Hence, the maximum number of circles in this scenario is C(10, 3) * C(14, 3) = 700 * 364 = 254800.

Scenario 3: Combination of collinear and non-collinear points
In this scenario, we consider a combination of collinear and non-collinear points. Let's assume we have 'a' collinear points and 'b' non-collinear points. To form a circle, we need 3 points. We can choose 3 points from the collinear set in C(a, 3) ways. For the remaining non-collinear points, we can choose 3 points in C(b, 3) ways. Hence, the maximum number of circles in this scenario is C(a, 3) * C(b, 3).

Optimization:
To maximize the number of circles, we need to find the scenario that yields the highest number of circles. Comparing the number of circles in each scenario, we find that Scenario 2 yields the maximum number of circles (254800).

Final Answer:
Hence, the maximum number of circles that can be drawn through any three points is 254800.