Comparing Triangles to Quadrilaterals!

Geometry Level 3

A set of 10 points lies in a plane such that no three points are collinear. Which quantity is greater A A or B B ?

Quantity A: The number of distinct triangles that can be created from the set.

Quantity B: The number of distinct quadrilaterals that can be created from the set.

The two quantities are equal. Quantity B is greater. The relationship can not be determined. Quantity A is greater.

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Hana Wehbi
Jun 9, 2017

The number of ways to create a triangle is 10 × 9 × 8 3 × 2 × 1 = 120 \frac{10\times9\times8}{3\times2\times1}= 120

While the number of ways to create a quadrilateral is 10 × 9 × 8 × 7 4 × 3 × 2 × 1 = 210 \frac{10\times9\times8\times7}{4\times3\times2\times1}= 210

Thus quantity B B is greater.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

You seem to be computing that the number of quadrilaterals as ( 10 4 ) \binom{10}{4} ; in other words, the number of ways of choosing four points.

It's not quite that simple. For example, suppose that in the four points you choose, one point lies inside the triangle formed by the other three points. Then there are 3 ways to form a (concave) quadrilateral from these four points. So the number of quadrilaterals is potentially greater than 210. (Of course, that means the answer is still Quantity B.)

Jon Haussmann - 3 years, 9 months ago

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Thank you for pointing it. I will give it a try for a better solution.

Hana Wehbi - 3 years, 9 months ago

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