What is the greatest number of points of intersection that can occur when $2$ different circles and $2$ different straight lines are drawn on the same piece of paper?
Make a diagram. Two geometric figures intersect if they have one or more points in common. Draw two circles which intersect in $2$ points. Draw a line which intersects the two circles in $4$ points. Draw another line which intersects the two circles in $4$ points and also intersects the first line. There are $\boxed{11}$ points of intersection. [asy]

draw(Circle((-0.7,0),1));
draw(Circle((0.7,0),1));

dot((0,0));

dot((0,0.7));
dot((0,-0.7));

draw((0,0)--(-2,0.6),Arrow);
draw((0,0)--(-2,-0.6),Arrow);
draw((0,0)--(2,0.6),Arrow);
draw((0,0)--(2,-0.6),Arrow);

dot((-1.58,0.47));
dot((-1.58,-0.47));
dot((1.58,0.47));
dot((1.58,-0.47));

dot((-0.29,0.08));
dot((-0.29,-0.08));
dot((0.29,0.08));
dot((0.29,-0.08));

[/asy]