Suppose that $20$ pillars of the same height have been erected along the boundary of circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is:

Each of the beams connects the top of two non-adjacent pillars, hence we have to find in how many ways we can select the required two pillars which are non-adjacent out of the given $20$ pillars.

As, ${}_{ }{}^{20}C_2$ is the number of beams connecting any two pillars whereas $20$ is the number of adjacent beams.

Thus, the total number of beams $={}_{ }{}^{20}C_2-20$

Using ${}^{n}C_r=\frac{n!}{r!\left(n-r\right)!},$

$=\frac{20!}{2!\times 18!}-20$

$=\frac{20\times 19\times 18!}{2!\times 18!}-20$

$=190-20$

$=170.$