There are three parallel straight lines. Two points A and B are marked on the first line, points C and D are marked on the second line, and points E and F are marked on the third line. Each of these six points can move to any position on its respective straight line. Consider the following statements:

I. The maximum number of triangles that can be drawn by joining these points is 18.
II. The minimum number of triangles that can be drawn by joining these points is zero.

Which of the statements given above is/are correct?

To determine the correct statements, let us evaluate the possible configurations of the 6 points:

• Statement I (Maximum Triangles): A triangle requires 3 non-collinear points. The total number of ways to choose 3 points from 6 is 6C3 = 20. Since the points can move freely to any position on their respective parallel lines, we can easily arrange them so that no three points are ever collinear. In this configuration, every set of 3 points forms a valid triangle. Thus, the maximum number of triangles is 20, making Statement I incorrect.
• Statement II (Minimum Triangles): To minimize the number of triangles, we must maximize collinear points. Because the points are freely movable, they can coincide. If points A and B coincide, C and D coincide, and E and F coincide, and we align all these locations along a single transversal line intersecting the three parallel lines, then all 6 points become collinear. When all points are collinear, no triangle can be formed. Thus, the minimum number of triangles is 0. Statement II is correct.

Therefore, only Statement II is correct.