Ten points lie on the circumference of a circle. Which of the following is the value that results when the number of triangles that can be created by connecting these points is subtrated from the number of heptagons (seven - sided polygons) that can be created by connected these points ?

Options

A. 210
B. 35
C. 21
D. 0

Answer

D

Explanation

An understandable but fruitless first reactio to this question is to draw a circle and then some triangles within it , before long, the lines become an indecipherable mass. There must be another solution. Think a bit more abstractly about what's happening mathematically when you take ten points and make triangles : you're essentially chunking a group of 10 into subgroups of 3. Similarly, when you make heptagons in this circle, you chunk subgroups of 7 from a group of 10. So the number of triangles is
`(n!)/(k!(n-k)!)`
`= (10!)/(3!7!)`
`= (10xx9xx8xx7xx6xx5xx4xx3xx3xx2xx1)/(3xx2xx1xx7xx6xx5xx4xx3xx2xx1)`
`= (10xx9xx8)/(3xx2xx1)`
`= 10xx3xx4=120`
And the number of heptangons is also `(10!)/(7!3!)`. No need to work out the value of this expression , it must have a value of 120, because it's equivalent to the expression representing the number of triangles. That the 3! and the 7! have been transposed makes no difference, since the order in which multiplication occurs is irrelevant. So the number of heptagons minus the number of triangles is `120-120=0`.