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

The correct option is 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 $={}^{11}{C}_3=165$

Number of circles formed out of 5 points by taking three points at a time $={}^5{C}_3=10$ It is given that 5 points lie on one circle.

$\therefore$ Required number of circles

= 165 - 10 + 1 = 156