Question

The number of triangles that can be formed with 10 points as vertices, n of them being collinear, is 110. Then n is
a) 3 b) 4 c) 5 d) 6; The number of triangles that can be formed with 10 points as vertices, n of them being collinear, is 110. Then n is; a) 3 b) 4 c) 5 d) 6

Answer 1

Please see the attachment

Answer 2

The number of points being collinear is 5.

• Total number of points on the plane is 10.

• The number of triangles that can be formed with the points on the plane where no three points are collinear is $^10C_3$.

• Now if there are some points on the plane that are collinear then the number of ways in which the triangle can be formed is

Number of triangles = Total number of Triangles - Selecting only three number of points that are collinear

• Now , there are n points on the plane that are collinear.

• Therefore,

Number of triangles = $^10C_3$ - $^nC_3$

• Total number of triangles so formed = 110.

• Now,

110 = 120 - $^nC_3$

$^nC_3$ = 10

• We can compare both sides , we obtain n as 5

n = 5

• The number of vertices that are collinear are 5.