I came across this question in my class:
There are 11 different points in the plane with no 3 points are on the same line.
a) How many circles do these points define? (Points define a circle if there is a unique circle through those points.)
b) How many circles would they define, if 4 points were on the same line?
I think, that we just need 2 points, to define a circle (one for the centre and 1 for the radius). In that case a) would be $11\times10=110$ different circles, however that seems to be incorrect.
How would you solve it?
Best Answer
It doesn't take two points to define a circle, because either of those two points could be the centre. Instead, the solution for part (a) is simply $\binom{11}3=165$ because three non-collinear points uniquely define a circle (assuming that multiplicity is counted, or that no four points are concyclic; if such a quartet existed the number of distinct circles would be lower).
For part (b), $\binom43=4$ circles become degenerate and need to be subtracted, so the answer is 161 circles in this case (again, with one of the assumptions above).