The number of triangles that can be formed by choosing vertices from a set of 12 points 7 of which lie on the same straight line is?

Question

Answer

Method-1

A triangle is formed on joining any three points out which at-least 2 are non co-linear points.

From a set of 12 points 3 can be selected in 12C3 = 220 ways

It is given that 7 points are co-linear. So, we can't get any triangle on selecting any 3 points from these 7 points. Selecting 3 points from 7 points can be done in 7C3 = 35 ways.

Therefore number of triangles formed = 220 - 35 = 185

Method-2

Given that there are 7 co-linear points so the remaining 5 points are non co-linear. To form a triangle the vertices must satisfy a condition that at-least 2 points must be non-col-linear. This can be achieved in the following ways,

- All three points are non-co-linear and non of them is from the 7 co-linear points set

=> 5C3 = 10 ways

- All three points are non co-linear and one of them is from the 7 co-linear points set

=> 5C2 x 7C1 = 70 ways

- Two points are co-linear and the other one is non co-linear with these two

=> 5C1 x 7C2 = 105 ways

Total number of triangles possible = 10 + 70 + 105 = 185

Answer

Number of triangles formed by choosing vertices from a set of 12 points with 7 points lying on the same line:

When 7 points lie on the same line, they cannot form a triangle with any other point. So, we need to consider the remaining 5 points to form triangles.

Case 1: No three points are collinear

In this case, any three points can be chosen to form a triangle. So, the number of triangles formed is:

C(5,3) = 10

Case 2: Three points are collinear

In this case, we can choose two points from the collinear points and one point from the remaining points to form a triangle. So, the number of triangles formed is:

C(7,2) x C(5,1) = 21 x 5 = 105

Case 3: Four points are collinear

In this case, we can choose two points from the collinear points and two points from the remaining points to form a triangle. So, the number of triangles formed is:

C(7,2) x C(5,2) = 21 x 10 = 210

Case 4: Five points are collinear

In this case, we can choose two points from the collinear points and one point from the remaining points. However, all such triangles will be degenerate triangles, i.e., they will have zero area. So, the number of triangles formed is:

0

Total number of triangles

The total number of triangles formed is the sum of the number of triangles formed in all the cases:

10 + 105 + 210 + 0 = 325

Explanation

When 7 points lie on the same line, they cannot form a triangle with any other point. So, we need to consider the remaining 5 points to form triangles. The number of triangles formed depends on the number of points that are collinear. We consider each case separately and calculate the number of triangles formed. We add the number of triangles formed in each case to get the total number of triangles.

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