Suppose that 20 pillars of the same height have been erected along...

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, $C2 20$ is the number of beams connecting any two pillars whereas $20$ is the number of adjacent beams.

Thus, the total number of beams $= C2 20 - 20$

Using $Cr n = \dfrac{n !}{r ! \left(n - r\right) !},$

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

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

$= 190 - 20$

$= 170 .$