1. What is the Triangle Counting Formula?

The formula calculates the number of triangles that can be formed by joining N points on a plane, where M of these points are collinear. It subtracts the invalid triangles formed by collinear points from the total possible triangles.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( C(n,3) \) — Number of ways to choose 3 points from n total points
• \( C(m,3) \) — Number of invalid triangles formed by m collinear points
• \( n \) — Total number of points
• \( m \) — Number of collinear points (must be ≤ n)

Explanation: The formula subtracts the triangles that cannot be formed due to collinearity from the total possible triangles.

3. Importance of Triangle Counting

Details: This calculation is fundamental in combinatorial geometry and has applications in computer graphics, pattern recognition, and mathematical problem solving.

4. Using the Calculator

Tips: Enter the total number of points (N) and the number of collinear points (M). Both values must be integers ≥ 3, and M must be ≤ N.

5. Frequently Asked Questions (FAQ)

Q1: Why subtract C(m,3) from C(n,3)?
A: Because collinear points cannot form valid triangles. C(m,3) represents the number of invalid triangle combinations from the collinear points.

Q2: What if all points are non-collinear?
A: If m = 0, then the formula simplifies to C(n,3), which gives the maximum number of triangles possible.

Q3: Can this formula handle multiple sets of collinear points?
A: This specific formula handles only one set of collinear points. For multiple sets, a more complex formula is needed.

Q4: What is the range of valid inputs?
A: Both n and m must be integers ≥ 3, and m must be ≤ n. Very large values may cause computational limitations.

Q5: How is the binomial coefficient calculated?
A: The calculator uses the multiplicative formula: C(n,k) = n×(n-1)×...×(n-k+1)/k! for efficient computation.