Question

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?

Answer 1

Four chosen chords form a bow-tie quadrilateral 2210 .

What is a quadrilateral?

• A quadrilateral is a polygon having four sides, four angles, and four vertices.
• The word 'quadrilateral' is derived from the Latin words 'quadri,' which means four, and 'latus', which means side.

• The above image is an example of a quadrilateral.
• The bowtie, or crossed rectangle, is a nonconvex semi-uniform quadrilateral with the same vertices as a rectangle but with two of the original sides removed and with the original's diagonals in place instead.

Given,

1.               We are aware that a quadrilateral requires four vertices (or points on the circle).
2. The cross can always be connected in one of two ways: horizontally or vertically.
3. My understanding of combinations is inadequate, but we can deduce that picking four points from a possible seven equals 21.
4. Four points can be used to create 21*2 = 42 "bow-tie quadrilaterals" on the circle by multiplying the two methods to join those lines (again, horizontally and vertically).
5. Since there are 2210 possible combinations of four chords out of a possible 21 chords, the probability is 42/2210, and we just need to reduce that fraction.

Therefore, Four chosen chords form a bow-tie quadrilateral 2210 .

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