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:

In problems involving collinear points, always subtract the combinations that form a straight line from the total combinations.

Answer 1

The Correct Option is B

We are given 12 points in total, of which 5 are collinear.

Step 1: Total ways to select 3 points from 12 points
The total number of ways to choose 3 points from 12 points is given by the combination formula: \[ \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]
Step 2: Subtract the number of ways to select 3 collinear points
Since no triangle can be formed by selecting 3 collinear points, we subtract the number of ways to choose 3 points from the 5 collinear points: \[ \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10 \] Therefore, the number of triangles that can be formed is: \[ 220 - 10 = 210 \] Thus, the number of triangles that can be formed is 210.