Q:

The number of circles that can be drawn out of $10$ points of which $7$ are co-linear

My attempt
The number of cases should be: 
$$10C2+3C2\cdot7C2+3C2\cdot7C1+3C3\cdot7C1+3C3\cdot7C2$$
But the answer is $85$. How?

A:

3 non-colinear points are needed to draw a circle.
GIVEN: 10 points are given of which 7 points are colinear.
There are 3 ways this can be done 
1) choosing three remaining points that are non co-linear = 3C3 = 1
2) choosing 2 points from the 3 remaining points and 1 point from the 7 colinear points = 3C2 * 7C1 = 3 * 7 = 21
3) choosing 1 point from the 3 remaining points and 2 points from the 7 colinear points = 3C1 * 7C2 = 3 * 21 = 63
Total = 63 + 21 + 1 = 85

