Given 11 points, of which 5 lie on one circle, other than these 5, no 4 lie on one circle.
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

Solution:

(b) 156

(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 $=165-10+1=156$

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