Question

There are $\mathrm{n}$ points in a plane of which $\mathrm{p}$ points are collinear. How many lines can be formed from these points?

• A. ${}^{\left(\mathrm{n}-\mathrm{p}\right)}\mathrm{C}_2$
• B. ${}^{\mathrm{n}}\mathrm{C}_2-{}^{\mathrm{P}}\mathrm{C}_2$
• C. ${}^{\mathrm{n}}\mathrm{C}_2-{}^{\mathrm{P}}\mathrm{C}_2+1$
• D. ${}^{\mathrm{n}}\mathrm{C}_2-{}^{\mathrm{P}}\mathrm{C}_2-1$

Answer 1

The correct option is C

${}^{\mathrm{n}}\mathrm{C}_2-{}^{\mathrm{P}}\mathrm{C}_2+1$

Explanation of the correct option

Given, there are $\mathrm{n}$ points in a plane and $\mathrm{p}$ number of points among them are collinear.

Number of lines formed out of $\mathrm{n}$ points $={}^{\mathrm{n}}\mathrm{C}_2$

Number lines formed out of $\mathrm{p}$ collinear points $={}^{\mathrm{p}}C_2$

Number of lines containing all $\mathrm{p}$ points $=1$

Therefore, the total number of lines formed $={}^{\mathrm{n}}\mathrm{C}_2-{}^{\mathrm{p}}\mathrm{C}_2+1$

Hence, option $\left(\mathrm{C}\right)$, ${}^{\mathrm{n}}\mathrm{C}_2-{}^{\mathrm{P}}\mathrm{C}_2+1$ is the correct answer.