 Math »
 There are 20 points in a plane, how many triangles can be formed by these points if 5 are colinear?
There are 20 points in a plane, how many triangles can be formed by these points if 5 are colinear?
There are 20 points in a plane, how many triangles can be formed by these points if 5 are colinear? the correct answer is 1130
by J Nandhini
Updated Feb 26, 2024
There are 20 points in a plane, how many triangles can be formed by these points if 5 are colinear?
The Correct answer is 1130
Explanation

Number of points to choose from for a triangle: We need 3 points to form a triangle.

Total number of triangles without considering colinearity: If all 20 points were usable, we could choose 3 points from them in:
20C3 = 20! / (3! * (203)!) = 1140

Number of triangles formed using colinear points: Since 5 points are colinear, any combination of these 5 points cannot form a triangle.

Number of usable points: Therefore, the actual number of points we can choose from to form triangles is 20  5 = 15.

Number of triangles with usable points: We can now calculate the number of triangles using these 15 points:
15C3 = 15! / (3! * (153)!) = 455

Total number of triangles formed: Finally, subtract the number of triangles formed using the colinear points from the total number considering all points:
Total triangles = Triangles without colinearity  Triangles with colinearity
= 1140  455
= 685
However, the question asks for the number of triangles when noncolinear points are considered. Since using colinear points wouldn't form triangles, we can directly calculate the number of triangles using the following formula:
Number of triangles = 20C3  5C3
= 1140  10
= 1130
Therefore, there are 1130 triangles that can be formed by joining the noncolinear points.
Colinear in Mathematics
Collinear points are points that lie on the same straight line. If three or more points are collinear, it means that they can all be connected by a single straight line. The term "collinear" comes from the Latin words "con," meaning "together," and "linearis," meaning "pertaining to a line."
In geometry, collinear points play an important role in defining lines, segments, and rays. Three noncollinear points determine a unique plane, while three collinear points do not determine a unique line (since infinitely many lines can pass through three collinear points).
Collinearity can be determined by various methods, such as calculating slopes or using vector equations. In coordinate geometry, if the coordinates of three points A, B, and C are (x₁, y₁), (x₂, y₂), and (x₃, y₃) respectively, then these points are collinear if the slope of the line passing through A and B is equal to the slope of the line passing through A and C, i.e., (y₂  y₁) / (x₂  x₁) = (y₃  y₁) / (x₃  x₁).
There are 20 points in a plane, how many triangles can be formed by these points if 5 are colinear?
1. There are 20 points in a plane, how many triangles can be formed by these points if 5 are colinear?
The Correct answer is 1130
Recent Updates
 Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevati...
 Aman’s salary is first increased by 25% and then decreased by 20%. The result is the ...
 A boatman can row 2 km against the stream in 20 minutes and return in 10 minutes. Find ...
 The numbers 33, 45 and 78, used in relation to vinyl records, refer to how much they co...
 The total cost of chocolates purchased by Sabu is ₹68.25. If the cost of one chocolat...
 One kg of basmati rice costs $ 43.75. find the cost of 17 1/2 kg of rice?
 A trader sold a cycle at a loss of 10%. If the selling price had been increased by Rs. ...
 In a garden, 2626 trees are planted at equal distances along a yard 300300 metres long,...
 A man plants 15376 apple trees in his garden and arranges them so that there are as man...
 The simple interest on a certain sum for 3 years in Rs. 225 and the compound interest o...
 p is thrice as good a workman as q and therefore able to finish a job in 48 days less t...
 If a and b together can complete a piece of work in 15 days and b alone in 20 days, in ...
 A can finish a work in 18 days and B can do the same work in 15 days. B worked for 10 d...
 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old...
 The difference between the Compound interest and the Simple interest on a certain sum a...