Question

Given 11 points, of which 5 lie on one circle, other than these 5, no 4 lie on one circle. Then the number of circles that can be drawn so that each contains at least 3 of the given points is
(a) 216
(b) 156
(c) 172
(d) none of these

Answer 1

(b) 156
We need at least three points to draw a circle that passes through them.
Now, number of circles formed out of 11 points by taking three points at a time = ¹¹C₃ = 165
Number of circles formed out of 5 points by taking three points at a time = ⁵C₃ = 10
It is given that 5 points lie on one circle.

$\therefore$ Required number of circles = 165$-$ 10 + 1 = 156