6 points = # triangles?

To form a triangle on the circle you will need to choose any 3 points on the circle and connect it. So it is the same as calculating the number of ways in which you can choose 3 points out of the total 6 points, which is $\binom{6}{3}$ .

$\binom{6}{3}=\frac{6!}{(3!)(3!)}=\frac{720}{6\times 6}=\boxed{20}$

Generalizing, the number of triangles formed by $n$ points on the circle is $\dfrac{n(n-1)(n-2)}{6}$

The number of triangles formed using $6$ points on a circle

$\frac{6!}{3! (6-3)!}= 20$