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:
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:
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When calculating the number of triangles, subtract the cases where all three points are collinear from the total number of combinations.
Updated On: Mar 25, 2026
- \(210\)
- \(200\)
- \(230\)
- \(220\)
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The Correct Option is A
Approach Solution - 1
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:
- 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:
- \( \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \)
- 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:
- \( \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10 \)
- 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 = \( 220 - 10 = 210 \)
- 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 \(\displaystyle (a)\).
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Approach Solution -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: \[ \binom{12}{3} - \binom{5}{3} = 220 - 10 = 210 \]
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: \[ \binom{12}{3} - \binom{5}{3} = 220 - 10 = 210 \]
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