Question

If four vertices of a regular octagon are chosen at random then the probability that the quadrilateral formed by them is a rectangle is

• A. $\frac{1}{8}$
• B. $\frac{2}{21}$
• C. $\frac{1}{32}$
• D. $\frac{1}{35}$

Answer 1

The correct option is D

$\frac{1}{35}$

Explanation for the correct option:

Step 1: Given information

In an octagon, there are 8 vertices, Out of which 4 vertices are chosen.

Hence, the total cases = ${}^8{\mathrm{C}}_4$

Apply the formula: $\begin{array}{rcl}{}^{\mathbf{n}}\mathbf{C}_{\mathbf{k}}& \mathbf{=}& \frac{\mathbf{n}\mathbf{!}}{\mathbf{k}\mathbf{!}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{k}\mathbf{)}\mathbf{!}}\end{array}$

$\begin{array}{rcl}\therefore {}^8\mathrm{C}_4& =& \frac{8!}{4!(8-4)!}\\ & =& \frac{8!}{4!\times 4!}\\ & =& \frac{1\times 2\times 3\times 4\times 5\times 6\times 7\times 8}{1\times 2\times 3\times 4\times 1\times 2\times 3\times 4}\\ & =& \frac{1680}{24}\\ \therefore {}^8\mathrm{C}_4& =& 70\end{array}$

If we choose 4 vertices, then the favorable outcome is 2 (i.e., the quadrilateral formed by 4 vertices are rectangles)

Step 2:Finding the required probability,

$$\begin{gathered} =\frac{2}{{}^8\mathrm{C}_4} \\ =\frac{2}{70} \\ =\frac{1}{35} \end{gathered}$$

Hence, the correct answer is an option(D).