Question

There are 10 straight lines in a plane, such that no 3 are concurrent and no 2 are parallel to each other. If points of intersection of above lines are joined, then maximum number of lines thus formed are

Answer 1

640

Answer 2

To find the maximum number of lines formed by joining the points of intersection of 10 straight lines, we follow these steps:

1. Calculate the number of intersection points:
   Given 10 straight lines in a plane, such that no 3 are concurrent and no 2 are parallel.
   Since no two lines are parallel, every pair of lines intersects at a unique point.
   Since no three lines are concurrent, all these intersection points are distinct.
   The number of intersection points is the number of ways to choose 2 lines out of 10, which is given by the combination formula $C(n, k) = \frac{n!}{k!(n-k)!}$.
   Number of intersection points = $C(10, 2) = \frac{10 \times 9}{2 \times 1} = 45$.

2. Calculate the total number of lines that can be formed by joining these intersection points:
   We have 45 distinct intersection points. A line is formed by choosing any two distinct points.
   The total number of ways to choose 2 points out of 45 is $C(45, 2)$.
   Total lines (initial count) = $C(45, 2) = \frac{45 \times 44}{2 \times 1} = 45 \times 22 = 990$.

3. Adjust for collinear points:
   The calculation in step 2 counts all possible lines. However, some of these intersection points are collinear, specifically those lying on one of the original 10 lines.
   Consider any one of the original 10 lines, say $L_1$. This line intersects the other 9 lines ($L_2, L_3, \dots, L_{10}$) at 9 distinct points. These 9 points are all collinear, as they lie on $L_1$.
   If we choose any 2 points from these 9 collinear points, they will form the line $L_1$.
   The number of ways to choose 2 points from these 9 collinear points is $C(9, 2) = \frac{9 \times 8}{2 \times 1} = 36$.
   These 36 pairs of points all define the same line ($L_1$). In our initial count of 990 lines, $L_1$ has been counted 36 times. But it is only 1 distinct line.
   So, for each of the 10 original lines, we have an overcount of $(C(9, 2) - 1) = (36 - 1) = 35$.
   Since there are 10 such original lines, the total overcount is $10 \times 35 = 350$.

4. Calculate the maximum number of distinct lines:
   To get the maximum number of distinct lines, we subtract the overcounted lines from the total initial count.
   Maximum number of lines = (Total lines from 45 points) - (Total overcount due to collinearity)
   Maximum number of lines = $990 - 350 = 640$.

The phrase "maximum number of lines thus formed" implies that no other set of three or more intersection points (other than those lying on the original 10 lines) are collinear.

Explanation of the solution:

1. Calculate total intersection points: $C(10, 2) = 45$.
2. Calculate total lines possible from these points: $C(45, 2) = 990$.
3. Identify collinear points: Each of the 10 original lines contains $10-1=9$ intersection points.
4. Calculate overcount: For each original line, $C(9, 2) = 36$ pairs of points define it, but it's only 1 line. So, each original line is overcounted by $36-1=35$.
5. Total overcount: $10 \times 35 = 350$.
6. Subtract overcount from total: $990 - 350 = 640$.