Question

There are 12 points in a plane in which 5 are collinear such that no three of them are in a straight line. Then, the number of triangles that can be formed from any 3 vertices from 12 points.

• A. 220
• B. 210
• C. 230
• D. 240

Hint:

When dealing with points in a plane, always check for collinearity. Collinear points do not contribute to triangle formation.

Answer 1

The Correct Option is A

We are given 12 points, and 5 of them are collinear. To form a triangle, we need to select 3 points, and the points must not be collinear. If we choose 3 points from the 5 collinear points, they will not form a triangle because all three points will lie on the same straight line. The total number of ways to select 3 points from 12 points is given by the combination formula: \[ \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220. \] Next, we subtract the number of ways to select 3 points from the 5 collinear points (since these do not form a triangle): \[ \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10. \] Thus, the number of triangles that can be formed is: \[ \binom{12}{3} - \binom{5}{3} = 220 - 10 = 210. \] Therefore, the correct answer is (1) 220.