Question

There are $\mathrm{m}$ points on a straight line $\mathrm{AB}$ and $\mathrm{n}$ points on another line $\mathrm{AC}$, none of them being the point $\mathrm{A}$. Triangles are formed from these points as vertices when

$\left(\mathrm{i}\right)$ $\mathrm{A}$ is excluded

$\left(\mathrm{ii}\right)$ $\mathrm{A}$ is included.

Then the ratio of the number of triangles in the two cases is?

• A. $\frac{\mathrm{m}+\mathrm{n}-2}{\mathrm{m}+\mathrm{n}}$
• B. $\frac{\mathrm{m}+\mathrm{n}-2}{2}$
• C. $\frac{\mathrm{m}+\mathrm{n}-2}{\mathrm{m}+\mathrm{n}+2}$
• D. None of these.

Answer 1

The correct option is A

$\frac{\mathrm{m}+\mathrm{n}-2}{\mathrm{m}+\mathrm{n}}$

Step $1:$ Evaluate the number of triangles if $\mathrm{A}$ is excluded

Given : There are $\mathrm{m}$ points on a straight line $\mathrm{AB}$ and $\mathrm{n}$ points on another line $\mathrm{AC}$, none of them being the point $\mathrm{A}$.

Thus, number of possible triangles$={}^{\mathrm{m}}\mathrm{C}_2\times {}^{\mathrm{n}}\mathrm{C}_1+{}^{\mathrm{n}}\mathrm{C}_2\times {}^{\mathrm{m}}\mathrm{C}_1$

$$\begin{gathered} =\frac{\mathrm{m}(\mathrm{m}-1)}{2}\times \mathrm{n}+\frac{\mathrm{n}(\mathrm{n}-1)}{2}\times \mathrm{m} \\ =\frac{\mathrm{mn}}{2}\left[\mathrm{m}-1+\mathrm{n}-1\right] \\ =\frac{\mathrm{mn}}{2}\left[\mathrm{m}+\mathrm{n}-2\right] \end{gathered}$$

Step $2:$ Evaluate the number of triangles if $\mathrm{A}$ is included

Given : There are $\mathrm{m}$ points on a straight line $\mathrm{AB}$ and $\mathrm{n}$ points on another line $\mathrm{AC}$, none of them being the point $\mathrm{A}$.

Thus, number of possible triangles$=1\times {}^{\mathrm{m}}\mathrm{C}_1\times {}^{\mathrm{n}}\mathrm{C}_1+\frac{\mathrm{mn}\left(\mathrm{m}+\mathrm{n}-2\right)}{2}$

$$\begin{gathered} =\mathrm{mn}\left[1+\frac{\mathrm{m}+\mathrm{n}-2}{2}\right] \\ =\frac{\mathrm{mn}(\mathrm{m}+\mathrm{n})}{2} \end{gathered}$$

Step $3:$ Evaluate the required ratio

Ratio$=\frac{{\displaystyle \frac{\mathrm{mn}}{2}}\left[\mathrm{m}+\mathrm{n}-2\right]}{{\displaystyle \frac{\mathrm{mn}(\mathrm{m}+\mathrm{n})}{2}}}$

Ratio$=\frac{\mathrm{m}+\mathrm{n}-2}{\mathrm{m}+\mathrm{n}}$

Hence, option $\left(\mathrm{A}\right)$, $\frac{\mathrm{m}+\mathrm{n}-2}{\mathrm{m}+\mathrm{n}}$ is the correct answer.