The number of triangles that are formed by choosing the vertices from a set of 12 point, seven of which lies on the same straight line is :a)105b)150c)175d)185Correct answer is option 'D'. Can you explain this answer?

Question

Answer

The given problem involves counting the number of triangles that can be formed by choosing vertices from a set of 12 points, out of which 7 lie on the same straight line.

To solve this problem, we can consider the different cases and combinations of vertices to form triangles.

1. No vertices on the same line:
In this case, we can choose any 3 vertices from the set of 12 without any restrictions. The number of ways to choose 3 vertices from 12 is given by the combination formula:
C(12, 3) = 12! / (3! * (12-3)!) = 220

2. Two vertices on the same line:
If we choose two vertices on the same line, the third vertex must be chosen from the remaining 5 points. The number of ways to choose 2 vertices on the same line is given by the combination formula:
C(7, 2) = 7! / (2! * (7-2)!) = 21

3. All three vertices on the same line:
If we choose all three vertices on the same line, there is only one possible way to form a triangle.

Now, we need to subtract the cases where all three vertices are on the same line, as they are not valid triangles.

Total number of triangles = (number of triangles without restrictions) - (number of triangles with all vertices on the same line)

= 220 - 1 = 219

Therefore, the correct answer is option 'D' (185).

Answer

We have points where 7 point are in line and other 5 points are non-coliinear so the triangles =

(As non-collinear points form triangle)

Similar Mathematics Doubts

There are three straight lines which are parallel and lie in the same plane A total number of 5 point on the first, 7 points on the second, mid 8 points on the third straight line as selected. The maximum number of triangles formed with the vertices at these points are

The number of numbers of four different digits that can be formed from the digits of the number 12 356 such that, the numbers are divisible by 4, is

The number of triangles that can be formed with 10 points as vertices, n of them being collinear, is 110. Then n is

In the figure, two 4-digit numbers are to be formed by filling the places with digits. The number of different ways in which the places can be filled by digits so that the sum of the numbers formed is also a 4-digit number and in no place the addition is with carrying is

The number of triangles whose vertices are at the Venices of an octagon but none of whose sides happen to come from the sides of the octagon is

Top Courses for Mathematics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams