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

Concept: A triangle needs 3 points that are not collinear. If all three points lie on the same straight line, they cannot form a triangle. So when the available points sit on two separate straight lines, every valid triangle must combine points from BOTH lines — specifically 2 points from one line and 1 point from the other (never all 3 from a single line).

Application:

1. Let the first line have n1 = 10 dots and the second line have n2 = 8 dots.

2. Case 1 — choose 2 dots from the first line and 1 dot from the second line: C(10,2) × C(8,1) = 45 × 8 = 360.

3. Case 2 — choose 1 dot from the first line and 2 dots from the second line: C(10,1) × C(8,2) = 10 × 28 = 280.

4. Add both cases together, since they are mutually exclusive ways of picking 3 non-collinear points: 360 + 280 = 640.

Cross-check: as an independent check, count all ways to choose any 3 points from the 18 dots combined, then remove the collinear (degenerate) selections: C(18,3) = 816 total triples; of these, C(10,3) = 120 have all three points on the first line and C(8,3) = 56 have all three on the second line (neither forms a triangle). Valid triangles = 816 − 120 − 56 = 640, confirming the result.