We define a bow-tie quadrilateral as a quadrilateral where two sides cross each other. An example of a bow-tie quadrilateral is shown below. Seven distinct points are chosen on a circle. We draw all $\binom{7}{2} = 21$ chords that connect two of these points. Four of these $21$ chords are selected at random. What is the probability that these four chosen chords form a bow-tie quadrilateral?

Step-by-step explanation:

Suppose we choose four points on the circle. Then there are exactly two ways that they can form the vertices of a bow-tie.

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Thus, there are exactly 2 times 7c4 = 70 bow-tie quadrilaterals that can be formed among the seven points.

There are 21 chords to choose from, so there are 21c4 = 5985 ways to choose four chords. Therefore, the probability that the four chords form a bow-tie quadrilateral is

70/5985 = 2/171.

this shows a reflection over the x-axis and is translated one unit down and 6 units to the right (x + 6, y - 1)

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