One line has 10 distinct dots and a second, separate line has 8 distinct dots,…

A triangle needs three points that do not all lie on one straight line. When the available points are split across two separate straight lines, a set of three points forms a triangle only if it draws points from both lines together — three points chosen entirely from a single line are always collinear and can never form a triangle.

1. The two lines carry 10 and 8 dots. A valid triangle can therefore be built in exactly two mutually exclusive ways: 2 vertices from the 10-dot line with 1 vertex from the 8-dot line, or 1 vertex from the 10-dot line with 2 vertices from the 8-dot line.

2. Number of ways for the first case (2 dots from the 10-dot line, 1 dot from the 8-dot line): 10C2 × 8C1 = 45 × 8 = 360.

3. Number of ways for the second case (1 dot from the 10-dot line, 2 dots from the 8-dot line): 10C1 × 8C2 = 10 × 28 = 280.

4. Adding the two mutually exclusive cases gives the total number of triangles: 360 + 280 = 640.

Cross-check: choosing any 3 points from all 18 dots in total gives 18C3 = 816 ways. Removing the two degenerate (collinear) selections — all three points on the 10-dot line (10C3 = 120 ways) or all three points on the 8-dot line (8C3 = 56 ways) — leaves 816 − 120 − 56 = 640, matching the direct case-by-case count.

Hence, the number of triangles that can be formed using these dots is 640.