CAT 2017 Slot 1 — QA Question 29
Let AB, CD, EF, GH, and JK be five diameters of a circle with center at O. In how many ways can three points be chosen out of A, B, C, D, E, F, G, H, J, K, and O so as to form a triangle?
Answer & solution
Answer: 160
There are 5 pairs of diametrically opposite points and the centre O.
If O is not selected, the number of triangles = 10C3 = 120.
If O is selected, the other two points can be selected in 10C2 - 5 = 40 ways. (when 3 points on a diameter are selected we will not get a triangle)
The number of triangles is = 120 + 40 = 160.
Alternately,
No. of ways of choosing 3 points out of given 11 points = 11C3 = 165.
Out of these 165 ways, 5 of these would give us 3 points on the diameters mentioned, which will not form a triangle.
Hence, no. of triangles = 165 - 5 = 160.
Hence, 160.