Four distinct [circles](https://artofproblemsolving.com/wiki/index.php/Circle) are drawn in a [plane](https://artofproblemsolving.com/wiki/index.php/Plane). What is the maximum number of points where at least two of the circles intersect?
$\mathrm{(A)}\ 8 \qquad\mathrm{(B)}\ 9 \qquad\mathrm{(C)}\ 10 \qquad\mathrm{(D)}\ 12 \qquad\mathrm{(E)}\ 16$
For any given pair of circles, they can intersect at most $2$ times. Since there are ${4\choose 2} = 6$ pairs of circles, the maximum number of possible intersections is $6 \cdot 2 = 12$. We can construct such a situation as below, so the answer is $\boxed{\mathrm{(D)}\ 12}$.
[2002 12B AMC-14.png](https://artofproblemsolving.com/wiki/index.php/File:2002_12B_AMC-14.png)