Ten points are marked on a straight line and eleven points are marked on…
2019
Ten points are marked on a straight line and eleven points are marked on another straight line. How many triangles can be constructed with vertices from among the above points?
Answer: C. 1045 — A triangle needs 3 vertices that are not all on the same straight line. Here all points lie on one of two lines: Line 1 has 10 points and Line 2 has 11…
- A.
495
- B.
550
- C.
1045
- D.
2475
Attempted by 9 students.
Show answer & explanation
Correct answer: C
A triangle needs 3 vertices that are not all on the same straight line. Here all points lie on one of two lines: Line 1 has 10 points and Line 2 has 11 points.
Any 3 points chosen from a single line are collinear and cannot form a triangle. So a valid triangle must use points from BOTH lines, in one of two ways:
Case 1 — 2 points from Line 1 and 1 point from Line 2: C(10, 2) × C(11, 1) = 45 × 11 = 495.
Case 2 — 1 point from Line 1 and 2 points from Line 2: C(10, 1) × C(11, 2) = 10 × 55 = 550.
Total triangles = 495 + 550 = 1045.
Cross-check: total ways to pick any 3 of the 21 points = C(21, 3) = 1330; subtract the collinear triples C(10, 3) + C(11, 3) = 120 + 165 = 285; 1330 − 285 = 1045. Both methods agree.