For the following two (02) items : There are 8 points on a plane out of which 4 points are collinear.
52
To solve this problem, we need to determine how many triangles can be formed by joining a set of given points. According to the information provided, there are 8 points on a plane, out of which 4 are collinear.
Let's break down the solution step by step:
- To form a triangle, we need to select 3 non-collinear points.
- First, find the total number of ways to select 3 points out of 8 points. This can be calculated using the combination formula: \(^{8}C_{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56\).
- Next, we need to subtract the number of triangles that cannot be formed because they fall on the same line. Since there are 4 collinear points, selecting any 3 of these collinear points will not form a triangle. The number of combinations of selecting 3 points from these 4 collinear points is: \(^{4}C_{3} = \frac{4 \times 3 \times 2}{3 \times 2 \times 1} = 4\).
- Therefore, the number of triangles that can be formed is the total combinations of 3 points from 8 points minus the collinear combination: \(56 - 4 = 52\).
Thus, the correct answer is 52. This is the total number of triangles that can be constructed from the given points, considering the collinear constraint.
Correct Option: 52
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