Question

There are n points in a plane, in which no three are in a straight line except ‘m’ which are all in a straight line. Find the number of different straight lines that can be formed with the given points as vertices.

Answer 1

Hint: This is a question based on permutations and combinations. Here we need to focus on the point that there is only one unique line passing through 2 given points. Also, remember that when any two of the three collinear points are joined they give the same line which passes through all the collinear points.

Complete step-by-step answer:
Let us first find the total number of lines that can be formed using the given n points. No matter whether the lines are overlapping or not. We know that there exists a single unique line passing through two given points. So, the number of lines is equal to the number of possible ways of selecting 2 points out of the given n points, i.e., $^{n}{{C}_{2}}$ .
Now out of these $^{n}{{C}_{2}}$ , the lies formed by taking 2 points out of the m collinear points gives overlapping lines. So, there are $^{m}{{C}_{2}}$ overlapping lines. So, the total number of distinct lines are:
Total number of lines-Number of overlapping lines+1
${{=}^{n}}{{C}_{2}}{{-}^{m}}{{C}_{2}}+1$
The term 1 is added because when we subtracted the overlapping lines we didn’t consider it one time even, but actually it is a possible case of line so it must be considered once.
Therefore, the answer to the above question is $^{n}{{C}_{2}}{{-}^{m}}{{C}_{2}}+1$ .

Note: In questions related to permutations and combinations, students generally face problems in deciding whether the question is based on selection or arrangement. Also, you might face problems in analysing whether to make cases or not. Also, be careful about the conditions mentioned in the question, as terms like non-parallel, non-collinear etc. are very crucial in such questions.