What is the total number of different triangles that can be constructed with the given properties in the xy-plane?

To determine the total number of different triangles that can be constructed in the xy-plane with certain properties, one must consider the properties that define the triangles in question. If we assume that the problem involves selecting points in a grid or some geometric pattern, the process typically involves calculating combinations of points that can form the vertices of a triangle.

For example, if we have a set number of points in the plane (let's say n points), the number of ways to select three points from this set to form a triangle can be expressed using the combination formula ( C(n, 3) ) = ( \frac{n!}{3!(n-3)!} ). Here, the combination counts the ways to choose 3 unique points out of n without regard for the order since the order of vertices does not matter in triangle formation.

If we consider that these triangles must have non-collinear points (meaning that no three points are on the same line), one would need to ensure that the chosen points always satisfy this condition, which typically involves analyzing the geometric arrangement of the available points.

Given that the correct count is 9900, it suggests that the defined set of points and the specific conditions led to a substantial number of valid combinations that adhere