+201 3. There are n different points on a circle. The number of triangles whose vertices are among the n points is positive, and equal to the number of hexagons whose vertices are among the n points. What is n?
+2407 Hello and welcome to web2.0calc :)))
(n)(n-1)(n-2)/3/2 is how many points of 3 you can select from n points (or number of triangles)
(n)(n-1)(n-2)(n-3)(n-4)(n-5)/6/5/4/3/2 is how many points of 6 you can choose from n points (number of hexagons)
(n)(n-1)(n-2)/3/2 = (n)(n-1)(n-2)(n-3)(n-4)(n-5)/6/5/4/3/2
1 = (n-3)(n-4)(n-5)/6/5/4
120 = (n-3)(n-4)(n-5)
n = 9
=^._.^=
+506 Since 6 points in a circle determine a hexagon, and 3 points determine a triangle:
\({n \choose 6} = {n \choose 3}\\ \frac{n!}{6!(n-6)!}=\frac{n!}{3!(n-3)!}\\ 6!(n-6)!=3!(n-3)!\\ 4\cdot5\cdot6=(n-3)(n-4)(n-5)\\ \)
Notice that the terms on both sides form an arithmetic progression. If you set a term on the RHS to the corresponding term on the LHS, you get (I used the smallest term as an example):
\(4=n-5\\\boxed{n=9}\)
oh no I got sniped
