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

Options

A. `(1)/(8)`
B. `(2)/(21)`
C. `(1)/(32)`
D. `(1)/(35)`

Explanation

To solve the problem of finding the probability that a quadrilateral formed by randomly selecting four vertices of a regular octagon is a rectangle, we can follow these steps:

### Step 1: Determine the total number of ways to choose 4 vertices from 8.
The number of ways to choose 4 vertices from 8 can be calculated using the combination formula:

\[ \text{Total ways} = \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} \]

### Step 2: Calculate the value of \(\binom{8}{4}\).
Calculating \(\binom{8}{4}\):

\[ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = \frac{1680}{24} = 70 \]

### Step 3: Determine the number of favorable outcomes (rectangles).
In a regular octagon, a rectangle can be formed by selecting two pairs of opposite vertices. Since there are 4 pairs of opposite vertices in an octagon, the number of ways to select pairs of opposite vertices is:

\[ \text{Favorable outcomes} = 2 \quad (\text{two rectangles can be formed}) \]

### Step 4: Calculate the probability.
The probability \(P\) that the quadrilateral formed by the selected vertices is a rectangle is given by the ratio of the number of favorable outcomes to the total number of outcomes:

\[ P = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{2}{70} = \frac{1}{35} \]

### Final Answer:
Thus, the probability that the quadrilateral formed by the four randomly chosen vertices of a regular octagon is a rectangle is:

\[ \boxed{\frac{1}{35}} \]