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

Given 12 points with 5 collinear, we aim to form triangles. A triangle requires 3 non-collinear points. Selecting 3 points from the 5 collinear ones will not form a triangle. The total combinations of selecting 3 points from 12 is:\[\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220.\]The combinations of selecting 3 points from the 5 collinear points, which do not form triangles, are:\[\binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10.\]The number of possible triangles is the total combinations minus the collinear combinations:\[\binom{12}{3} - \binom{5}{3} = 220 - 10 = 210.\]Therefore, 210 triangles can be formed.