One line has 10 dots, and a second line has 8 dots. How many triangles can be…

Concept: A triangle needs 3 points that are NOT collinear. Here, all points lie only on two straight lines, so 3 chosen points are collinear (and cannot form a triangle) only when all 3 come from the same line. Every other split of the 3 points across the two lines gives a valid triangle.

Method 1 — total minus collinear cases:

1. There are 10 + 8 = 18 dots in all, so the number of ways to pick any 3 of them is 18C3 = 816.

2. Picks where all 3 points are on the 10-dot line are collinear and invalid: 10C3 = 120.

3. Picks where all 3 points are on the 8-dot line are collinear and invalid: 8C3 = 56.

4. Removing both invalid (collinear) cases from the total: 816 − 120 − 56 = 640.

Cross-check — direct valid-case count:

A valid triangle must split its 3 points across the two lines — either 2 dots from the 10-dot line with 1 from the 8-dot line, or 1 dot from the 10-dot line with 2 from the 8-dot line:

1. 2 from the 10-dot line and 1 from the 8-dot line: 10C2 × 8C1 = 45 × 8 = 360.

2. 1 from the 10-dot line and 2 from the 8-dot line: 10C1 × 8C2 = 10 × 28 = 280.

3. Adding the two valid patterns: 360 + 280 = 640 — matching Method 1.

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