To determine the number of triangles that can be formed by selecting three points from the given grid of dots, let's analyze the setup.

Step-by-Step Solution

1. Identify Points and Configuration:

   • The diagram shows a grid of dots in a 4x4 arrangement, resulting in 16 points.

2. Selecting Combinations:

   • To form a triangle, we need to select three points from these 16. The number of ways to choose 3 points out of 16 is given by the combination formula: $\binom{16}{3} = \frac{16!}{3!(16 - 3)!} = \frac{16 \times 15 \times 14}{6} = 560$
   • This gives us the total number of possible combinations of three points.

3. Filtering Collinear Combinations:

   • Not all combinations will result in a triangle. If three points are collinear, they cannot form a triangle.
   • In a 4x4 grid, collinear sets include horizontal lines, vertical lines, and some diagonal lines.

4. Count Collinear Sets:

   • Horizontal and Vertical Lines: Each row and each column has 4 points. We can select 3 points out of these 4 in each row or column.
     • There are 4 rows and 4 columns, each providing $\binom{4}{3} = 4$ collinear combinations.
     • Total collinear combinations for rows and columns: $4 \times 4 + 4 \times 4 = 32$.
   • Diagonals: We also need to consider the diagonals:
     • There are two main diagonals with 4 points each, giving $2 \times \binom{4}{3} = 8$ collinear combinations.
     • Shorter diagonals contain fewer than 3 points and do not affect our count.

5. Calculate Non-Collinear Combinations:

   • Total combinations of three points: 560
   • Total collinear combinations: 32 (rows/columns) + 8 (diagonals) = 40
   • Therefore, the number of triangles that can be formed is: $560 - 40 = 520$

Answer

The number of triangles that can be formed by selecting three points from the grid is 520.

Would you like further details or have any questions?

Here are 5 related questions that expand on this concept:

1. How would the answer change if the grid size were increased to a 5x5 arrangement?
2. What if we considered only equilateral triangles? How many could we form?
3. How do we approach the problem if the points are arranged in a circular pattern instead of a grid?
4. How can we extend this method to find quadrilaterals instead of triangles?
5. What are the mathematical principles behind identifying collinear points in any arrangement?

Tip: When analyzing point arrangements, look for patterns of symmetry and regularity to simplify counting distinct geometric shapes.