Question

the number of straight lines which can be drawn through 90 points given that 10 of them lie on a straight line​

Answer 1

Answer:

Example 1 There are 10 points in a plane, no 3 of which are collinear. How many different lines can be formed by joining these points?

Solution To form a line, all we need to do is select any 2 points (subtask 1), and then join them (subtask 2). The number of ways to select any points (out of 10 distinct points) will be 10C2. Once we select the points, there is only 1 straight line which will be formed using these points. Therefore the number of lines will be 10C2 x 1 or 45. Here’s a crazy figure to illustrate.

permutation combination examples

Example 2 Suppose 4 of the 10 points in the previous example were collinear, and no three of the remaining are collinear. Find the number of straight lines which can be formed. Also find the number of triangles which can be formed.

Solution Since 4 particular points are now collinear, there will be only 1 line which will be formed by joining any 2 of these 4, instead of 4C2 if they had been non collinear. So, we’ve lost 4C2 or 6 lines, and instead we got only 1. Therefore, the total number of lines will be 10C2 – 4C2 + 1 = 40

There is one more way in which we can calculate the number of lines. Let’s divide the points in two groups: the collinear group of 4 points, and the non-collinear group of 6 points.

To count the number of lines, we have three possible cases. First, the lines formed using the points of the collinear group – only 1 line. Second, the lines formed using only the points of the non-collinear group – 6C2 or 15 lines. Third, select one point from the collinear group (4C1 ways) and the other from the non-collinear group (6C1 ways) – the number of such lines will be 4C1 x 6C1 = 24.