If four distinct lines are drawn in a plane, which of the following cannot be the number of distinct points of intersection of these lines?

Introduction / Context:
This problem is about combinatorial geometry and tests your understanding of how many intersection points can be formed by a given number of lines in a plane. Such questions are common in aptitude tests to check logical thinking, imagination and basic combinatorics, rather than heavy computation.

Given Data / Assumptions:
• We have exactly four distinct lines in a plane. • Lines may intersect, be parallel or even concurrent (many passing through one point). • We are asked which given number cannot represent the count of distinct intersection points.

Concept / Approach:
For n distinct lines in a plane, if no two are parallel and no three are concurrent, the maximum possible number of intersection points is nC2, because each pair of lines intersects in exactly one point. For n = 4, the maximum is 4C2 = 6. By adjusting whether some lines are parallel or concurrent, we can obtain fewer intersection points. Any suggested number greater than 6 would be impossible. Additionally, we should reason whether smaller numbers like 0, 4 or 5 can be constructed by choosing a suitable arrangement of the lines.

Step-by-Step Solution:
1. For four lines, compute the maximum intersections if all pairs meet: 4C2 = (4 * 3) / 2 = 6. 2. Therefore, there can never be more than 6 distinct intersection points. 3. Check if 0 is possible: take all four lines parallel to each other. Then no two lines meet, so 0 intersections is possible. 4. Check if 4 is possible: take three concurrent lines (all passing through a single point) and a fourth line that intersects each of these three at different points. We then have 3 + 1 = 4 distinct intersection points. 5. Check if 5 is possible: take three lines in general position (forming 3 intersections), and a fourth line parallel to one of the three but intersecting the other two. This gives 3 existing intersections plus 2 new ones, for a total of 5. 6. Finally, consider 7: this exceeds the maximum possible 6, so 7 intersections is impossible with only four lines.

Verification / Alternative check:
You can visualise different configurations or quickly sketch them. For maximum intersections, draw four lines so that no two are parallel and no three pass through the same point; you will count exactly six intersection points. This confirms that 6 is the absolute maximum. Any number less than or equal to 6 may or may not be possible, so we must construct examples or reason as above, but any number greater than 6 is automatically impossible.

Why Other Options Are Wrong:
• 0: Possible when all four lines are mutually parallel. • 5: Possible with three lines in general position and a fourth line parallel to one and intersecting the other two. • 4: Possible with three concurrent lines plus a fourth line intersecting them at distinct points. • 6: This is actually the maximum possible and is achievable when all pairs of lines intersect uniquely.

Common Pitfalls:
A common error is to think that if four lines are in general position they always produce 4 intersections, which is incorrect. Another mistake is to ignore the possibility of all lines being parallel or many lines being concurrent. Remember to use the formula nC2 for the maximum intersection points and then think creatively about arrangements that reduce the count.

Final Answer:
The number of intersection points that cannot occur is 7.