Question

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is:

• A. \(210\)
• B. \(200\)
• C. \(230\)
• D. \(220\)

Hint:

When calculating the number of triangles, subtract the cases where all three points are collinear from the total number of combinations.

Answer 1

The Correct Option is A

The problem requires calculating the number of triangles that can be formed using 12 points on a plane, with a special condition that 5 of these points are collinear. Here's how to solve it step-by-step:

1. To find the total number of triangles that can be formed, we start by selecting any 3 points from the 12 points. The number of ways to choose 3 points from 12 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of points and \( r \) is the number of points to choose:
   • \( \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \)
2. However, 5 of these points are collinear. A set of collinear points does not form a triangle because they all lie on a single straight line. Therefore, we need to subtract the number of ways to choose 3 points from these 5 collinear points:
   • \( \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10 \)
3. Therefore, the total number of triangles that can be formed by the 12 points, considering the collinear constraint, is:
   • Total triangles = Total combinations of 3 points - Collinear combinations
   • Total triangles = \( 220 - 10 = 210 \)
4. Thus, the total number of triangles that can be formed with the given constraints is \(210\).

Therefore, the correct answer is \(210\), which is option \(\displaystyle (a)\).

Answer 2

The total number of ways to choose 3 points out of 12 is given by the combination formula: \[ \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \] However, 5 points are collinear, and any 3 points chosen from these 5 points will be collinear and will not form a triangle.
The number of ways to choose 3 points out of 5 collinear points is: \[ \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10 \] So, the total number of triangles that can be formed is: \[ \binom{12}{3} - \binom{5}{3} = 220 - 10 = 210 \]