How many straight lines can be drawn with the help of 12 points where 5 of them are collinear?

General
1

how many straight lines can be drawn with the help of 12 points where 5 of them are collinear?

2

To determine the number of straight lines that can be drawn with the help of 12 points, where 5 of them are collinear, we need to consider the different combinations of points that can form a straight line.

Let’s break down the problem:

  • There are a total of 12 points.
  • 5 of these points are collinear, which means they lie on the same line.
  • The remaining 7 points are not collinear with the first 5 points.

Now, let’s consider the possibilities:

  1. Drawing a line using all 12 points: In this case, we can form one straight line using all 12 points.

  2. Drawing a line using only the 5 collinear points: Since these points are already on the same line, we can form one straight line using just these 5 points.

  3. Drawing a line using 4 collinear points and 1 point from the remaining 7 points: We can choose 4 collinear points out of the 5, which can be done in 5C4 ways (5 choose 4). For each combination of 4 collinear points, we can choose 1 point from the remaining 7 points, giving us a total of 5C4 * 7 lines.

  4. Drawing a line using 3 collinear points and 2 points from the remaining 7 points: We can choose 3 collinear points out of the 5 in 5C3 ways. For each combination of 3 collinear points, we can choose 2 points from the remaining 7 points, resulting in 5C3 * 7C2 lines.

  5. Drawing a line using 2 collinear points and 3 points from the remaining 7 points: We can choose 2 collinear points out of the 5 in 5C2 ways. For each combination of 2 collinear points, we can choose 3 points from the remaining 7 points, giving us 5C2 * 7C3 lines.

  6. Drawing a line using 1 collinear point and 4 points from the remaining 7 points: We can choose 1 collinear point out of the 5 in 5C1 ways. For each collinear point, we can choose 4 points from the remaining 7 points, resulting in 5C1 * 7C4 lines.

  7. Drawing a line using only 1 point from the remaining 7 points: We can choose 1 point from the remaining 7 points in 7C1 ways, giving us 7C1 lines.

To determine the total number of lines, we sum up the lines from each of these possibilities:

Total lines = 1 + 1 + (5C4 * 7) + (5C3 * 7C2) + (5C2 * 7C3) + (5C1 * 7C4) + 7C1

By evaluating these combinations, the total number of straight lines that can be drawn with the help of 12 points, where 5 of them are collinear, is determined.