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]
11
Level 5
Prealgebra
Problem: 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?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the greatest number of points of intersection, we need to analyze the interactions between the circles and the lines. 

First, consider the intersections between the two circles. Two different circles can intersect at most at two points. Therefore, the maximum number of intersection points from the circles is 2.

Next, we analyze the intersections between the two lines. Two different straight lines can intersect at most at one point. Thus, the maximum number of intersection points from the lines is 1.

Now, we need to consider the intersections between the circles and the lines. Each line can intersect a circle at most at two points. Since there are two lines and two circles, we calculate the intersections as follows:

- The first line can intersect the first circle at 2 points.
- The first line can intersect the second circle at 2 points.
- The second line can intersect the first circle at 2 points.
- The second line can intersect the second circle at 2 points.

This gives us a total of intersections from the lines and circles as follows:
2 (first line with first circle) + 2 (first line with second circle) + 2 (second line with first circle) + 2 (second line with second circle) = 8 intersection points.

Now, we combine all the intersection points calculated:
- 2 points from the intersection of the circles,
- 1 point from the intersection of the lines,
- 8 points from the intersections of the lines with the circles.

Adding these together gives:
2 (circle-circle) + 1 (line-line) + 8 (line-circle) = 11.

Thus, the greatest number of points of intersection that can occur when 2 different circles and 2 different straight lines are drawn is 11.<|END_OF_LATENT|> The final answer is $11$.