Question

Four points are randomly chosen from the vertices of a regular 12-sided polygon. If is the probability that the four chosen points form a rectangle (including squares), then find p.

Answer 1

Answer:

The age of a father is 3 times the age of his son. 5 years ago father was five

times as old as his son was . Find their present age.The age of a father is 3 times the age of his son. 5 years ago father was five

times as old as his son was . Find their present age.

The age of a father is 3 times the age of his son. 5 years ago father was five

times as old as his son was . Find their present age.

Step-by-step explanation:

The age of a father is 3 times the age of his son. 5 years ago father was five

times as old as his son was . Find their present age.

Answer 2

Concept Introduction:

The Combination formula in mathematics illustrates how many different methods there are to acquire a specified sample of "k" items from a bigger collection of "n" distinct objects. As a result, we have a Combination when the order is irrelevant, and a Permutation when the order is significant.

Given: Four points are randomly chosen from the vertices of a regular $12$-sided polygon.

To Find:

We have to find the value of,  probability that the four chosen points form a rectangle.

Solution:

According to the problem,

$3$  squares and $6+6$ non-squares, making $15$ rectangles

Each non-square can be rotated $30$∘ to produce a different rectangle, but $180$∘ produces the original version. So six purple possibilities and six red possibilities.

Each square can be rotated $30$∘ to produce a different square, but $90$∘ produces the original version. So three green possibilities.

Now, choose the first vertex, that can be done in $12$ ways(assuming all the vertices are distinct).

So, in order to find diagonally opposite vertex, you can have only one way(because each vertex has unique diametrically opposite vertex in an even sided regular polygon), so, in one way.

Now, you have to find another vertex. As, only $10$ points are left, you can do this in $10$ ways. And its diametrically opposite vertex in 1 way.

So, the number of way is

$12*1*10*1=120$

And, here I forgot to remove the repeating. In this form of counting, each rectangle is counted $8$ times. So divide the above number by $8$. So, the ultimate result is

$120/8=15$.

∴ Probability is $\frac{15}{120} = 1/8$

Final Answer:

The value of P is $1/8$.

#SPJ3