Question

Four points are randomly chosen from the vertices
1
of a regular 12-sided polygon. If is the
р
probability that the four chosen points form a
rectangle (including squares), then find p.​

Answer 1

Ok, I think you tried to form rectangles. And, I am assuming all the vertices are distinct. So when you are forming a rectangle, it is tantamount to finding the number of ways you can draw its diagonals.

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 1 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 repeatings. In this form of counting, each rectangle is counted 8 times. So divide the above number by 8. So, the ultimate result is

1208=15..